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Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem Hindawi Publishing Corporation Advances in Astronomy Volume 2015, Article ID 615029, 7 pages http://dx.doi.org/10.1155/2015/615029 Research Article Stability of the Moons Orbits in Solar System in the Restricted Three-Body Problem Sergey V. Ershkov Institute for Time Nature Explorations, M.V. Lomonosov’s Moscow State University, Leninskie Gory 1-12, Moscow 119991, Russia Correspondence should be addressed to Sergey V. Ershkov; [email protected] 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. 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 3 around the 2nd of two giant-bodies 1, 2 (which are rotating around their common centre of masses on Kepler’s trajectories), themassofwhichisassumedtobelessthanthemassofcentralbody.UnderassumptionsofR3BP,weobtaintheequationsofmotion which describe the relative mutual motion of the centre of mass of 2nd giant-body 2 (planet) and the centre of mass of 3rd body (moon) with additional effective mass ⋅2 placed in that centre of mass ( ⋅2 +3),where is the dimensionless dynamical parameter. They should be rotating around their common centre of masses on Kepler’s elliptic orbits. For negligible effective mass ( ⋅2 +3) it gives the equations of motion which should describe a quasi-elliptic orbit of 3rd body (moon) around the 2nd body 2 (planet) for most of the moons of the planets in Solar System. 1. Introduction of restricted three-body problem (such as infinitesimal masses and negligible deviations of the main orbital elements). The stability of the motion of the moon is the ancient problem Nevertheless, KAM also is known to assume the appro- which leading scientists have been trying to solve during last priate Hamilton formalism in proof of the central KAM- 400 years. A new derivation to estimate such a problem from theorem [1]; the dynamical system is assumed to be Hamil- a point of view of relative motions in restricted three-body ton’s system and all the mathematical operations over such a problem (R3BP) is proposed here. dynamicalsystemareassumedtobeassociatedwithaproper Systematic approach to the problem above was suggested Hamilton system. earlier in KAM- (Kolmogorov-Arnold-Moser-) theory [1]in According to the Bruns theorem [5], there are no other which the central KAM-theorem is known to be applied for invariants except well-known 10 integrals for three-body pro- researches of stability of Solar System in terms of restricted blem (including integral of energy and momentum); this is a classical example of Hamilton’s system. But in case of restri- three-body problem [2–5], especially if we consider photogra- cted three-body problem, there are no other invariants except vitational restricted three-body problem [6–8] with addi- only one, Jacobian-type integral of motion [3]. tional influence of Yarkovsky effectofnongravitationalnature Such a contradiction is the main paradox of KAM-theory; [9]. it adopts all the restrictions of restricted three-body problem, KAM is the theory of stability of dynamical systems [1] but nevertheless it proves to use the Hamilton formalism, which should solve a very specific question in regard to which assumes the conservation of all other invariants (the the stability of orbits of so-called “small bodies” in Solar integral of energy, momentum, etc.). System, in terms of restricted three-body problem [3]: indeed, To avoid ambiguity, let us consider a relative motion in dynamics of all the planets is assumed to satisfy restrictions three-body problem [2]. 2 Advances in Astronomy 2. Equations of Motion Analysing system (3) we should note that if we sum all the aboveequationsonetoeachotheritwouldleadustothe Let us consider the system of ODE for restricted three-body result below: problem in barycentric Cartesian coordinate system, at given initial conditions [2, 3]: R1,2 + R2,3 + R3,1 = 0. (4) (q − q ) (q − q ) q =−{ 1 2 1 2 + 1 3 1 3 }, 1 1 3 3 q1 − q2 q1 − q3 If we also sum all the equalities (∗) one to each other, we should obtain (q − q ) (q − q ) q =−{ 2 1 2 1 + 2 3 2 3 }, 2 2 3 3 (1) q2 − q1 q2 − q3 R1,2 + R2,3 + R3,1 = 0. (∗∗) (q − q ) (q − q ) q =−{ 3 1 3 1 + 3 2 3 2 }, 3 3 3 3 Under assumption of restricted three-body problem, we q − q q − q 3 1 3 2 assume that the mass of small 3rd body 3 ≪1,2,respec- tively; besides, for the case of moving of small 3rd body 3 as where q1, q2,andq3 mean the radius vectors of bodies 1, a moon around the 2nd body 2, let us additionally assume 2,and3,respectively; is the gravitational constant. |R2,3|≪|R1,2|. The system above could be represented for relative motion So taking into consideration (∗∗),weobtainfromsystem of three bodies as shown below (by the proper linear transfor- (3) the following: mations): (q1 − q2) R (q − q ) +( + ) 1,2 1 2 1 2 3 R +( + ) = , q − q 1,2 1 2 3 0 1 2 R1,2 (q − q ) (q − q ) R = { 3 1 + 2 3 }, R +( + ) 2,3 3 3 3 2,3 2 3 3 q − q q − q R 3 1 2 3 2,3 (5) (q − q ) R (R + R ) (q − q ) +( + ) 2 3 1,2 1,2 2,3 2 3 2 3 3 = { − }, q − q 1 3 3 2 3 R1,2 R1,2 + R2,3 (2) (q − q ) (q − q ) R + R + R =0, = { 3 1 + 1 2 }, 1,2 2,3 3,1 1 3 3 q3 − q1 q1 − q2 (q − q ) where the 1st equation of (5) describes the relative motion of 2 (q − q ) +( + ) 3 1 massive bodies (which are rotating around their common cen- 3 1 1 3 3 q3 − q1 tre of masses on Kepler’s trajectories); the 2nd describes the orbitofsmall3rdbody3 (moon) relative to the 2nd body 2 (q − q ) (q − q ) = { 1 2 + 2 3 }. (planet), for which we could obtain according to the trigono- 2 3 3 metric “Law of Cosines” [10]: q1 − q2 q2 − q3 Let us designate the following: R R +( + ) 2,3 2,3 2 3 3 R1,2 =(q1 − q2), R2,3 R2,3 =(q2 − q3), (∗) 1 R2,3 + (1 + 3cos ) R2,3 3 R (6) R1,2 1,2 R3,1 =(q3 − q1). R Using of (∗) above, let us transform the previous system to 1 2,3 ≅−3cos( R1,2) , another form: 3 R R1,2 1,2 R R R R +( + ) 1,2 = { 3,1 + 2,3 }, 1,2 1 2 3 3 3 3 where is the angle between the radius-vectors R2,3 and R1,2. R1,2 R3,1 R2,3 Equation (6) could be simplified under the additional |R |≪|R | R2,3 R1,2 R3,1 assumption above 2,3 1,2 for restricted mutual R +( + ) = { + }, 2,3 2 3 3 1 3 3 (3) motions of bodies 1 and 2 in R3BP [3]asbelow: R2,3 R1,2 R3,1 R R R ( + ) R +( + ) 3,1 = { 2,3 + 1,2 }. R +( 2 3 + 1 )⋅R = . 3,1 1 3 3 2 3 3 2,3 3 3 2,3 0 (7) R3,1 R2,3 R1,2 R2,3 R1,2 Advances in Astronomy 3 Moreover, if we present (7) in the form below third-body gravity), they are roughly equivalent, but the pro- posed ansatz is obviously an alternative approach, which R couldbemoreeffectivefortheinvestigationsofmutualrela- R +( ++)⋅ ⋅ 2,3 = , 2,3 1 2 3 0 R tive motion and stability of the moons orbits in Solar System. 2,3 If the total sum of dimensionless parameters ( + ) is (8) 3 negligible then (8) should describe a stable quasi-circle orbit 1 R2,3 3 =( ⋅ ), =( ), of 3rd body (moon) around the 2nd body 2 (planet). Let us 3 2 R1,2 2 consider the proper examples which deviate (differ) to some extent from the negligibility case ( + ) →0 above (Table 1) then (8) describes the relative motion of the centre of mass [11]: of 2nd giant-body 2 (planet) and the centre of mass of 3rd (1) Nereid-Neptune: body (moon) with the effective mass ( ⋅2 +3),whichare −6 rotating around their common centre of masses on the stable ( + ) = (35.81 + 0,29) ⋅ 10 , eccentricity 0.7507. (9) Kepler’s elliptic trajectories. Besides, if the dimensionless parameters , →0,then (8) should describe a quasi-circle motion of 3rd body (moon) (2) Triton-Neptune: around the 2nd body 2 (planet). −6 ( + ) = (0.01 + 210) ⋅ 10 , (10) 3. The Comparison of the Moons in eccentricity 0.000016. Solar System (3) Iapetus-Saturn: As we can see from (8), is the key parameter which deter- (+) = (54.46 + . ) ⋅ −6, . minesthecharacterofmovingofthesmall3rdbody3 (the 3 4 10 eccentricity 0 0286 (11) moon)relativetothe2ndbody2 (planet). Let us compare such a parameter for all considerable known cases of orbital (4) Titan-Saturn: moving of the moons in Solar System [11](Table 1). −6 ( + ) = (2.2 + 240) ⋅ 10 , eccentricity 0.0288. (12) 4. Discussion (5) Io-Jupiter: As we can see from Table 1, the dimensionless key parameter ( + ) = ( . + ) ⋅ −6, . , which determines the character of moving of the small 0 168 47 10 eccentricity 0 0041 (13) 3rd body 3 (moon) relative to the 2nd body 2 (planet), is varying for all variety of the moons of the planets (in (6) Callisto-Jupiter: −6 Solar System) from the meaning 0.0004 ⋅ 10 (for Proteus of −6 ( + ) = ( .
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