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An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem. (English) Zbl 1157.70005

Summary: The model of extended Sitnikov Problem contains two equally heavy bodies of mass \(m\) moving on two symmetrical orbits w.r.t. the centre of gravity. A third body of equal mass \(m\) moves along a line \(z\) perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the primaries plane the third body performs an oscillatory motion around it. The motion of the three bodies is described by a coupled system of second-order differential equations for the radial distance of the primaries \(r\) and the third mass oscillation \(z\). This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non-integrable and therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence of the function iteration method is given, and the iterations are carried out to order 1 in \(r\) and to order 2 in \(z\). For small values of the initial oscillation amplitude of the third mass, we obtain results in very good agreement to numerically obtained solutions.

MSC:

70F07 Three-body problems

Software:

Mathematica
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References:

[1] Dvorak R., Sun Y.S.: The phase space structure of the extended Sitnikov-problem. Celest. Mech. Dyn. Astron. 67, 87–106 (1997). doi: 10.1023/A:1008265310911 · Zbl 0912.70008
[2] Eddington, A.S.: Relativitätstheorie in mathematischer Behandlung, Springer Verlag, Berlin, p. 40 (1925) · JFM 51.0696.04
[3] Faruque S.B.: Solution of the Sitnikov problem. Celest. Mech. Dyn. Astron. 87, 353–369 (2003). doi: 10.1023/B:CELE.0000006721.86255.3e · Zbl 1106.70323
[4] Hagel J.: A new analytic approach to the Sitnikov problem. Celest. Mech. Dyn. Astron. 53, 267–292 (1992). doi: 10.1007/BF00052614 · Zbl 0757.70006
[5] Hagel J., Lhotka C.: A high order perturbation analysis of the Sitnikov problem. Celest. Mech. Dyn. Astron. 93, 201–228 (2005). doi: 10.1007/s10569-005-0521-1 · Zbl 1129.70007
[6] Liu J., Sun Y.S.: On the Sitnikov Problem. Celest. Mech. Dyn. Astron. 49, 285–302 (1990). doi: 10.1007/BF00049419 · Zbl 0718.70005
[7] MacMillan, W.D.: An integrable case in the restricted problem of three bodies, A.J. 27, 11–13 (1913)
[8] Moser J.: Stable and random motion in dynamical systems. Ann. Math. Stud. 77, 83–99 (1973)
[9] Nayfeh A.H.: Perturbation Methods. Wiley, London, Sidney, Toronto (1973)
[10] Perdios E.A.: The manifolds of families of 3D periodic orbits associated to Sitnikov motions in the restricted three-body problem. Celest. Mech. Dyn. Astron. 99, 85–104 (2007). doi: 10.1007/s10569-007-9088-3 · Zbl 1162.70311
[11] Press W.H. et al.: Numerical Recipes in FORTRAN Second Edition. Cambridge University press, New York (1986)
[12] Soulis P.S., Bountis T., Dvorak R.: Stability of motion in the Sitnikov 3-body problem. Celest. Mech. Dyn. Astron. 99, 129–148 (2007). doi: 10.1007/s10569-007-9093-6 · Zbl 1162.70312
[13] Soulis P.S., Papadakis T., Bountis T.: Periodic orbits and bifurcations in the Sitnikov four-body problem. Celest. Mech. Dyn. Astron. 100, 251–266 (2008). doi: 10.1007/s10569-008-9118-9 · Zbl 1254.70029
[14] Völkl, B.: Numerische und analytische Untersuchungen zum erweiterten Sitnikov-Problem, Diplomarbeit, Universitätssternwarte Wien, 2007
[15] Wolfram S.: The Mathematica Book, 5th edn. Wolfram Media/Cambridge University Press, Cambridge (2003)
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