Advances in Astronomy

Volume 2015, Article ID 615029, 7 pages

http://dx.doi.org/10.1155/2015/615029

## Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem

Institute for Time Nature Explorations, M.V. Lomonosov’s Moscow State University, Leninskie Gory 1-12, Moscow 119991, Russia

Received 1 April 2015; Revised 3 June 2015; Accepted 4 June 2015

Academic Editor: Elmetwally Elabbasy

Copyright © 2015 Sergey V. Ershkov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the equations of motion of three-body problem in a *Lagrange form* (which means a consideration of relative motions of 3 bodies in regard to each other). Analyzing such a system of equations, we consider in detail the case of moon’s motion of negligible mass around the 2nd of two giant-bodies , (*which are rotating around their common centre of masses on Kepler’s trajectories*), the mass of which is assumed to be less than the mass of central body. Under assumptions of R3BP, we obtain the equations of motion which describe the relative mutual motion of the centre of mass of 2nd giant-body (planet) and the centre of mass of 3rd body (moon) with additional effective mass placed in that centre of mass , where *ξ* is the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler’s elliptic orbits.
For negligible effective mass it gives the equations of motion which should describe a *quasi-elliptic* orbit of 3rd body (moon) around the 2nd body (planet) for most of the moons of the planets in Solar System.

#### 1. Introduction

The stability of the motion of the moon is the ancient problem which leading scientists have been trying to solve during last 400 years. A new derivation to estimate such a problem from a point of view of relative motions in restricted three-body problem (R3BP) is proposed here.

Systematic approach to the problem above was suggested earlier in KAM- (*Kolmogorov*-*Arnold*-*Moser*-) theory [1] in which the central KAM-theorem is known to be applied for researches of stability of Solar System in terms of* restricted* three-body problem [2–5], especially if we consider* photogravitational* restricted three-body problem [6–8] with additional influence of* Yarkovsky* effect of nongravitational nature [9].

KAM is the theory of stability of dynamical systems [1] which should solve a very specific question in regard to the stability of orbits of so-called “small bodies” in Solar System, in terms of* restricted* three-body problem [3]: indeed, dynamics of all the planets is assumed to satisfy restrictions of* restricted* three-body problem (*such as infinitesimal masses and negligible deviations of the main orbital elements*).

Nevertheless, KAM also is known to assume the appropriate Hamilton formalism in proof of the central KAM-theorem [1]; the dynamical system is assumed to be* Hamilton’s* system and all the mathematical operations over such a dynamical system are assumed to be associated with a proper Hamilton system.

According to the Bruns theorem [5], there are no other invariants except well-known 10 integrals for three-body problem (*including integral of energy and momentum*); this is a classical example of Hamilton’s system. But in case of* restricted* three-body problem, there are no other invariants except only one, Jacobian-type integral of motion [3].

Such a contradiction is the main paradox of KAM-theory; it adopts all the restrictions of* restricted* three-body problem, but nevertheless it proves to use the Hamilton formalism, which assumes the conservation of all other invariants (*the integral of energy, momentum, etc.*).

To avoid ambiguity, let us consider a relative motion in three-body problem [2].

#### 2. Equations of Motion

Let us consider the system of ODE for restricted three-body problem in barycentric Cartesian coordinate system, at given initial conditions [2, 3]:where , , and mean the radius vectors of bodies , , and , respectively; is the gravitational constant.

The system above could be represented for relative motion of three bodies as shown below (by the proper linear transformations):Let us designate the following:Using of above, let us transform the previous system to another form:Analysing system (3) we should note that if we sum all the above equations one to each other it would lead us to the result below:If we also sum all the equalities one to each other, we should obtainUnder assumption of restricted three-body problem, we assume that the mass of small 3rd body , respectively; besides, for the case of moving of small 3rd body as a moon around the 2nd body , let us additionally assume .

So taking into consideration , we obtain from system (3) the following:where the 1st equation of (5) describes the relative motion of 2 massive bodies (*which are rotating around their common centre of masses on Kepler’s trajectories*); the 2nd describes the orbit of small 3rd body (moon) relative to the 2nd body (planet), for which we could obtain according to the trigonometric “Law of Cosines” [10]:where *α* is the angle between the radius-vectors and .

Equation (6) could be simplified under the additional assumption above for* restricted* mutual motions of bodies and in R3BP [3] as below:Moreover, if we present (7) in the form belowthen (8) describes the relative motion of the centre of mass of 2nd giant-body (planet) and the centre of mass of 3rd body (moon) with the effective mass , which are rotating around their common centre of masses on the stable Kepler’s elliptic trajectories.

Besides, if the dimensionless parameters , , then (8) should describe a quasi-circle motion of 3rd body (moon) around the 2nd body (planet).

#### 3. The Comparison of the Moons in Solar System

As we can see from (8), is the key parameter which determines the character of moving of the small 3rd body (the moon) relative to the 2nd body (planet). Let us compare such a parameter for all considerable known cases of orbital moving of the moons in Solar System [11] (Table 1).