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Periodic solution of the nonlinear Sitnikov restricted three-body problem
Periodic solution of the nonlinear Sitnikov restricted three-body problem
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Periodic solution of the nonlinear Sitnikov restricted three-body problem Elbaz I. Abouelmagd, Juan Luis Garc´ıa Guirao, Ashok Kumar Pal PII: DOI: Reference:
S1384-1076(19)30223-4 https://doi.org/10.1016/j.newast.2019.101319 NEASPA 101319
To appear in:
New Astronomy
Received date: Revised date: Accepted date:
20 July 2019 19 August 2019 15 September 2019
Please cite this article as: Elbaz I. Abouelmagd, Juan Luis Garc´ıa Guirao, Ashok Kumar Pal, Periodic solution of the nonlinear Sitnikov restricted three-body problem, New Astronomy (2019), doi: https://doi.org/10.1016/j.newast.2019.101319
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Highlights • The properties of Sitnikov motion are analyzed. • The multiple scales method is used to remove secular terms. • The periodic solution of the circular Sitnikov problem is found. • Comparisons among a numerical solution and the solutions provide by the multiple scales method are investigated.
1
Periodic solution of the nonlinear Sitnikov restricted three-body problem Elbaz I. Abouelmagda,b , Juan Luis Garc´ıa Guiraoc , Ashok Kumar Pald a Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan – 11421, Cairo, Egypt. b Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia c Departamento de Matem´ atica Aplicada y Estad´ıstica. Universidad Polit´ ecnica de Cartagena, Hospital de Marina, 30203-Cartagena, Regi´ on de Murcia, Spain d Department of Mathematics and Statistics, School of Basic Science, Manipal University Jaipur, Jaipur – 303007, Rajasthan, India
Abstract The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time. Keywords: Sitnikov restricted three–body problem; Multiple scales method; Periodic solution. 1. Introduction One of the most important dynamical systems in celestial mechanics and space mechanics is the three–body problem. It is according to variation of its versions and applications in either stellar dynamics or astrodynamics. The general model of this problem is the three–body problem in which three bodies move in three dimensions under their mutual gravitational interactions, within frame Newton’s laws of motion and universal gravitation, without imposed constraints on the initial positions and velocities vectors of these bodies. An extended and a comprehensive review on the three–body problem is stated by [1, 2]. Although the general three–body problem is common in the field of celestial mechanics, there is no big significant contributions in its study at the literature on the properties of this system and in particular for the periodic Email addresses: [email protected] or [email protected] (Elbaz I. Abouelmagd), [email protected] (Juan Luis Garc´ıa Guirao), [email protected]. (Ashok Kumar Pal) Preprint submitted to New Astronomy
solutions. Note that there were only three families of periodic orbits which have been found until 2013 within frame of zero angular momentum and equal masses, since this problem “(general three–body problem)” was recognized [3]. But in the recent years with the serious work of Xiaoming Li and his team, new results and many families of Newtonian periodic orbits related to three–body problem were found, see for instance [4–6]. We underline that 695 families of Newtonian periodic collision less orbits within frame of equal masses in the three bodies system, with zero angular momentum were found numerically in [4]. Also in the case of two masses equal and the third one different, 1349 new families of periodic orbits have been constructed in [5]. None of these new families were found before, except the famous Figure– Eight family. In other words, 1223 among of 1349 families are entirely new. Furthermore within of frame three–body problem in [6], using a clean numerical simulation strategy for chaotic dynamic systems, 316 collision less periodic orbits are obtained with a few chosen values for the parameter of mass ratio. Authors remarks that 313 collision less September 17, 2019
free-fall periodic orbits obtained are new.
tion of the infinitesimal body are developed in dimensionless variable, such that depends on time t and the eccentricity parameter e of the primaries. The problem is considered a generalization to a “MacMillan problem” which is a special case from the elliptical Sitnikov problem when e = 0. In the case of e 6= 0 the Sitnikov model represents a chaotic system and admits many different kinds of motions, whoever it has a simple form than the ones found in chaotic dynamical systems. One of the most important properties of this problem is its mathematical simple structure, but at the same time it has a complexity on its possible type of motion. These properties make the Sitnikov model a unique system the field of stellar dynamics.
Within framework of imposed restrictions on the third body, where it has negligible mass with respect to the other two bodies and moving under the gravitational effects of these two bodies, the three–body problem will be reduced to the restricted problem, in this case the third body is called the infinitesimal body and the others bodies are called the primaries. With more restrictions, if the primaries motion is in the same plane with circular or elliptical orbits, we will get the spatial circular or elliptical restricted three–body problem in which the third body moves in three dimensional space [7]. But in the case of the infinitesimal body and the primaries move in the same plane, we will obtain the most familiar dynamical system in celestial mechanics, which is called the “planar restricted three–body problem”.
[26] studied on the stability and bifurcation of vertical motion into three-dimensional periodic orbits families in the Sitnikov problem within frame of restricted N–body model. [27] studied the circular Sitnikov model by alternation of stability and instability within frame of the family of periodic vertical motions when the varying continuous of monotonic amplitude is considered. [22] computed the solution by the Lindstedt–Poincar´e method and compared with existing solutions. [28] have computed analytically the families of symmetric periodic orbits in the elliptic Sitnikov problem. They also provide a qualitative information on the bifurcation diagram of such families of periodic orbits. [29] studied the photogravitational Sitnikov problem in the restricted three–body problem with oblateness. [24] provided the existence criteria of odd periodic solutions with a prescribed number of zeros.
Within frame of planar circular model, an analytical investigations on the effects of some perturbed forces such that non–sphericity of the primaries in the locations of equilibrium points, and their linear stability, as well as periodic orbits around these points have been developed by [8–11]. Some works also are devoted to investigate the locations of collinear points and the motion in the vicinity of such points, see for instance [12, 13]. The numerical integration of this problem using Lie series is also analyzed in these works [14, 15]. An important work for extending the planar circular restricted three–body to more realistic and effective models in the field of celestial mechanics are addressed by [16–18].
This paper includes eight sections, it is organized in the following way: A brief overview is stated on literature review due to periodic solutions of general three–body problem and Sitnikov problem in Section 1. In Section 2 the model description of the three–body problem to Sitnikov model is addressed. In Section 3, the dynamics characteristics of motion are investigated. In Section 4, the general structure of multiple scales method is stated. The first and second approximated solutions via multiple scales method and the periodicity conditions are constructed and explained in Sections 5 and 6. In addition the numerical results and a comparisons among the obtained solutions are investigated in Section 7. Finally the conclusion are stated in Section 8.
One of the reduced models of the three–body problem is the Sitnikov problem, which is considered a sub–case from the spatial elliptic restricted three–body problem, where the infinitesimal body (third body) has oscillation motion along Z−axis, which is perpendicular to the orbital plane of the primaries. In this model the primaries have equal masses and rotate in elliptical orbits. This model is called elliptical Sitnikov problem, that was constructed by [19]. The case when the primaries move in circular orbits around their common center of mass, is called the circular Sitnikov problem, which was stated by [20]. The Sitnikov models are considered the simplest sub– cases of general N −−body problem which can be used as a first approximation in many cases of real situations for astronomical problems. The existence of oscillation motion for this problem is proved for first time by [21], thereby it is called Sitnikov problem. It has also been studied by several researcher such as [22–24]. The Newton–Raphson basins of convergence for the equilibrium points in this problem are also explored numerically by [25]. These basins are studied under the sphericity lack of primaries. They also revealed the attracting regions with several types of two dimensional planes, by using the classical Newton–Raphson iterative method.
2. Model description Let m1 , m2 and m the three masses of the bodies b1 , b2 and b3 where b1 and b2 are the primaries, while b is the third body. One of the versions of the three–body problem is the spatial circular restricted three–body problem. In which the primaries move in the same plane in circular orbits around their barycenter with the normalized angular velocity to its motion plane. The motion of third body, which called the infinitesimal body is free in order to move in the 3−dimensional space, and its motion does not affect
In the elliptical Sitnikov problem, the equation of mo3
the primaries motion. We also assume that the total mass of the primaries is one, in which m1 = 1 − µ and m2 = µ.
Eqs.(1, 2, 3) can be reduced to the circular Sitnikov problem or the motion of the test particle along the vertical Z−axis d V (z), (4) z¨ = dz where 1 V (z) = p . (5) 1/4 + z 2
By choosing a synodic coordinates system with the origin at the center of mass, as well as the primaries b1 and b2 are fixed on the X−axis at (−µ, 0, 0) and (1 − µ, 0, 0) respectively. Now we assume that r, r1 and r2 are the positions vectors of the infinitesimal body with respect to the origin of the synodic coordinates and the primaries. Furthermore the scale of the time is chosen, so that the gravitational constant is one. Then the equations of motion of the infinitesimal body will be controlled by x ¨ − 2y˙ = Wx , y¨ + 2x˙ = Wy ,
3. Properties of Sitnikov motion The motion of a test particle in Sitnikov model is governed by a nonlinear differential equation as in Eq.(4), where the only exact analytical solution can be found in terms of elliptic functions, which is not easy in its mathematical structures. Thereby the alternative possible solutions with mathematical forms for this equation are an approximated solutions, which reflect the original features of the exact solution. These solutions can be developed by using the perturbation techniques. Although the perturbation techniques are considered effective tools in obtaining a solution of nonlinear differential equations, but these techniques have their own limitations. Because they depend on a very small physical parameter, which does not appear in Eq.(4). In fact the most applications of perturbation methods are not applied without the existence of this parameter.
(1)
z¨ = Wz , where W (x, y, z) = and r= r1 = r2 = here
p p
p
µ 1−µ 1 2 (x + y 2 ) + + , 2 r1 r2
(2)
x2 + y 2 + z 2 , (x + µ)2 + y 2 + z 2 ,
(3)
(x + µ − 1)2 + y 2 + z 2 ,
r = |r| , r1 = |r1 | , and r2 = |r2 |.
It is clear that from the Fig.2, the potential is symmetric around z = 0, which means that the potential may represent a harmonic oscillator motion.
The spatial restricted three–body problem is not only a special model from three–body problem, but also the Sitnikov problem, which can be considered it as a special model from the general problem or from the spatial restricted problem, when the primaries are equal in mass and the infinitesimal body moves only onpZ−axis. Hence x = y = 0 , µ = 1/2 and r1 = r2 = 1/4 + z 2 . See Fig.1 for the configuration of this model. In this context,
V(z)
-0.5
-1.0
-1.5
-2.0 -4
-2
0
2
4
z Figure 2: The curve of the potential V (z) (Color figure online).
Let E be the total mechanical energy, then 1 2 1 z˙ − p = E. 2 1/4 + z 2
(6)
The region of motion can be investigated with a simple analysis of Eq.(6). Since the potential value V (z) ∈ [−2, 0) for all values of z. Hence the motion will undefined in the rang E < −2 and bounded for −2 < E < 0, while
Figure 1: The configuration of Sitnikov problem (Color figure online).
4
it will be unbounded in the rang E > 0. The region of motion in 3–dimensional and its phase portrait are drawn in Fig. 3 and Fig. 4. In phase portrait figure, the regions
bounded and the third body will also enforced to remain at the center of masses. This is clear from the phase portrait in Fig. 3 when the value of total energy (E) is tends to (−2). Within frame of the total energy is negative and it is close to −2, then the third body will be near and close to the center of masses, thereby |z| 1. Then with a help of Eqs.(4, 5) the equation of motion up O(z 3 ) can be written as z¨ + ω 2 z − $2 z 3 = 0, (7) where ω 2 = 8 and $2 = 48. To find the periodic solution of Eq.(7), an approximation may still be found by introducing a very small parameter into the problem, but in the end its value will replace by one. This method is acceptable when the terms recalling the differential equation are quasi-linear or small in of themselves. Parameters introduced in this way are always called “Place Keeping Parameters“, see for more details [30]. Now we interchange each z by δz where δ 1, and take ε = δ 2 , of course, also ε 1, then Eq.(7) will take the following form
Figure 3: The region of possible motion in the 3–dimensional when E < 0 (Color figure online).
2
z¨ + ω 2 z − ε$2 z 3 = 0.
0.5
(8)
Then Eq.(8) represents equation of motion in standard form to apply one of the perturbation methods, like multiple scales method or Lindstedt–Poincar´e technique.
1 0
dz/dt
4. Multiple scales solution 0
Let the initial conditions of motion be z(0) = Z, z(0) ˙ = 0.
-1
(9)
0
0
To continue in our procedures, we will write the solution z(t) in the form of a perturbation series: 1
z(t) =
-2 -2
-1
0
1
2
∞ X
εn zn (t, τ, σ),
(10)
n=0
z Figure 4: The phase portrait of motion for different values of total energy E (Color figure online).
where τ = εt and σ = ε2 t.
of periodic motion are determined by closed contours when −2 < E < 0, and the region of unbounded motion will be determined when E ≥ 0.
In fact, the provided solution by multiple scales method includes the fast and slow variables. Although these variables are dependent, the processing strategy of multiple scales method for removing the secular terms is handling the two slow variables as independent variables. Hence the ordinary derivative with respect to the fast variable t will be expanded to the differential operator Dt . Which includes the partial derivatives with respect to the fast variable t and the slow variables τ and σ. With the help of the chain rule, the differential operator Dt is defined as
Furthermore the third body will has zero velocity (z˙ = 0) in three cases. The first two cases when it tends to infinity (z → ±∞) with zero energy (E = 0), then there are two stationary points at positive (negative) infinity. In these two case the motion is considered unbounded and the third body will be enforced to remain stationary at infinity. While the third case when total energy tends to −2 (E = −2) and the infinitesimal body moves to stay at the common center of masses (z = 0), then the motion is
∂ ∂ ∂ d = Dt := ( + ε + ε2 ), dt ∂t ∂τ ∂σ 5
(11)
hence
∞ X d z(t) = Dt εn zn (t, τ, σ). dt n=0
with the initial conditions Z0 (0, 0) + Z¯0 (0, 0) = Z,
(12)
With the conditions that zn (t, τ, σ) is continuous and differentiable with respect to t, τ and σ, and substituting Eq.(11) into Eq.(12) with some simple calculations, one obtains ∂z0 ∂z1 ∂z0 dz = +ε + dt ∂t ∂t ∂τ (13) ∂z2 ∂z1 ∂z0 2 3 +ε + + + O(ε ), ∂t ∂τ ∂σ 2 ∂ 2 z0 ∂ z1 ∂ 2 z0 d2 z = + ε + 2 dt2 ∂t2 ∂t2 ∂τ ∂t 2 ∂ z2 ∂ 2 z1 (14) +2 ∂t2 ∂τ ∂t 3 2 + O(ε ). +ε ∂ 2 z0 ∂ 2 z0 + +2 ∂σ∂t ∂τ 2 Using Eqs.(10, 13, 14) and Eq.(8), with the help of Eq.(9) and equalling the coefficients which has the same order in ε, we will get a series of linear partial differential equations which govern the functions of zi (t, τ, σ), for i = 0, 1, 2, these equations are ∂ 2 z0 = −ω 2 z0 , ∂t2 z0 (0, 0, 0) = Z,
∂ z0 (0, 0, 0) = 0. ∂t
∂ 2 z1 ∂ 2 z0 2 3 = −ω z + 48z − 2 , 1 0 ∂t2 ∂τ ∂t z1 (0, 0, 0) = 0, ∂ ∂ z1 (0, 0, 0) = − z0 (0, 0, 0). ∂t ∂τ
Here Z0 (τ, σ) is an arbitrary complex function in the slow variable τ , σ and Z¯0 (τ, σ) is its complex conjugate. 5. First approximated solution Insert Eq.(18) into Eq.(16), one obtains ∂ 2 z1 = − w2 z1 ∂t2 + 48 Z03 (τ, σ)ei3ωt + Z¯03 (τ, σ)e−i3ωt
(20)
+ s1 (τ, σ)eiωt + s¯1 (τ, σ)e−iωt ,
where
∂Z0 s1 (τ, σ) = 2 72Z02 Z¯0 − iω , ∂τ ∂ Z¯0 . s¯1 (τ, σ) = 2 72Z¯02 Z0 + iω ∂τ
(21) (22)
It is clear that the solution of Eq.(20) will contain a secular term, if the coefficients of the functions Exp (iωt) and Exp (−iωt) have non zero values. Because these functions are the solutions of the homogeneous equation associated to Eq.(20). In order to avoid the appearance of these secular terms in the solutions of Eq.(20) with respect to the fast variable t, the coefficients s1 (τ, σ) and s¯1 (τ, σ) have to equal zero.
(15)
5.1. Periodicity condition of first approximation Equalling coefficients s1 (τ, σ) and s¯1 (τ, σ) to zero not only to avoid the secular terms but also to determine the function Z0 (τ, σ) and Z¯0 (τ, σ) with an elegant way, but s1 (τ, σ) and s¯1 (τ, σ) are just two conjugate quantities. It is enough to solve Eq.(21) with s1 (τ, σ) = 0, because the solution of Eq.(22) for s¯1 (τ, σ) = 0 will give the same results for the solution of Eq.(21). This processing will give the warranty that the solution of z1 (t, τ, σ) will not involve secular terms.
(16)
∂ 2 z2 = −ω 2 z2 + 144z02 z1 ∂t2 ∂ 2 z1 ∂ 2 z0 ∂ 2 z0 −2 −2 − , (17) ∂τ ∂t ∂σ∂t ∂τ 2 z2 (0, 0, 0) = 0, ∂ ∂ ∂ z2 (0, 0, 0) = − z1 (0, 0, 0) − z0 (0, 0, 0). ∂t ∂τ ∂σ
To carry out our purpose, we will solve Eq.(21) with the help of the polar coordinates (R, θ). Hence we suppose that Z0 (τ, σ) = R(τ, σ)eiθ(τ,σ) ,
To accomplish our steps, we have to find the solutions of Eqs.(15–17) without including secular terms, specifically, the solutions are periodic functions in the fast variable t. This means that we have to determine the formulas of the functions zi (t, τ, σ) where zi (t + 2π, τ, σ) = zi (t, τ, σ). Now we will assume that the solution of Eq.(15) in the independently handled variables t, τ and σ is z0 (t, τ, σ) = Z0 (τ, σ)eiωt + Z¯0 (τ, σ)e−iωt ,
(19)
Z0 (0, 0) − Z¯0 (0, 0)) = 0.
(23)
R, θ : R → R,
insert Eq.(23) into Eq.(21), and use the condition of s1 (τ, σ) = 0, one obtains
Z0 (τ, σ) = α(σ)e
(18) Z¯0 (τ, σ) = α(σ)e 6
if (σ)−
72α2 (σ) τ ω
−if (σ)−
2
,
72α (σ) τ ω
(24) ,
s¯2 (τ, σ) = 288α2 (σ)Z¯1 (τ, σ) + 2iω
where α(σ) and f (σ) are arbitrary functions in the slow variable σ, with
+ 144α2 (σ)Z1 (τ, σ)e−2iθ(τ,σ) dα(σ) iω 2 ω + 144iτ α2 (σ) 2e−iθ(τ,σ) dσ + ω2 df (σ) +3024α5 (σ) + ω 3 α(σ) dσ
R(τ, σ) = α(σ), (25)
2
72α (σ) θ(τ, σ) = f (σ) − τ. ω
Substituting Eqs. (24),(25) into Eq.(18), then we get the first approximated periodic solution in form i h (26) z0 (t, τ, σ) = α(σ) eiγ(t,τ,σ) + e−iγ(t,τ,σ) , (27)
Now Eq.(20) can be written as ∂ 2 z1 = −w2 z1 Z03 (τ, σ)ei3ωt + Z¯03 (τ, σ)e−i3ωt , ∂t2 and its general solution is
Z1 (0, 0) + Z¯1 (0, 0) =
here i=
√
(29)
6. Second approximated solution Substituting Eq.(26) and Eq.(28) into Eq.(17), then the solution of z2 (t, τ, σ) is controlled by ∂ 2 z2 = − w2 z2 ∂t2 864α5 (σ) i5γ(t,τ,σ) −i5γ(t,τ,σ) + e + e ω2 6048α5 (σ) i3γ(t,τ,σ) −i3γ(t,τ,σ) (30) e + e − ω2 " # Z1 (τ, σ)ei(ωt+2θ(τ,σ)) + + 144α2 (σ) Z¯1 (τ, σ)e−i(ωt+2θ(τ,σ))
Let us go back to periodicity condition s2 (τ, σ) = 0, which leads to a system of partial differential equations, and there is extra difficulty to find its analytical solution in most cases. This is considered the major defect of multiple scales method. In order to avoid this difficulty, we have to note that we search for an approximated analytical solution of the second approximated (z1 ), we mean that the solution does not general or exact solution. Hence the particular solution of proper selection of the arbitrary functions which make a convenient with the initial conditions may be acceptable, if its validation is tested.
+ s2 (τ, σ)eiωt + s¯2 (τ, σ)e−iωt , where ∂Z1 (τ, σ) ∂τ
+ 144α2 (σ)Z¯1 (τ, σ)e2iθ(τ,σ) (31) 2 dα(σ) 2 iθ(τ,σ) iω ω − 144iτ α (σ) 2e dσ − , 2 df (σ) ω −3024α5 (σ) − ω 3 α(σ) dσ
.
One has to note that we arrange the equation of z1 , within frame its solution has no secular terms or is not resonance with the solution of homogenous part to find the arbitrary functions Z0 (τ, σ) and Z¯0 (τ, σ). The same processing will be done to find the arbitrary functions Z1 (τ, σ) and Z¯1 (τ, σ), within frame the solution of z2 has no secular terms. That will be repeated n times to get the periodic function, which represents the solution of zn .
−1, α0 = α(0) and f0 = f (0).
s2 (τ, σ) = 288α2 (σ)Z1 (τ, σ) − 2iω
(32)
6.1. Periodicity condition of second approximation The periodicity condition of obtaining periodic solution for the second approximation is available through the solution of Eq.(31) with s2 (τ, σ) = 0 and Eq.(32) with s¯2 (τ, σ) = 0. But it is also enough to solve Eq.(31) with s2 (τ, σ) = 0. Because the output solutions of s2 (τ, σ) = 0 and s¯2 (τ, σ) = 0 will be the same. This processing will also give the warranty that the solution of z2 will not include secular or unbounded terms and at least there are no secularities appear in the solution of perturbation series up to the third term. Furthermore we can construct the arbitrary functions Z1 (τ, σ) and Z¯1 (τ, σ).
z1 (t, τ, σ) = Z1 (τ, σ)eiωt + Z¯1 (τ, σ)e−iωt , i (28) 6α3 (σ) h i3γ(t,τ,σ) −i3γ(t,τ,σ) e + e − , ω2 with the initial conditions 12 3 α cos 3f0 , ω2 0 36α03 Z1 (0, 0) − Z¯1 (0, 0) = i [4 sin f0 + sin 3f0 ] , ω2
As a similar way in the first approximated solution, the solution of Eq.(30) will contain secular terms, if the coefficients s2 (τ, σ) and s¯2 (τ, σ) of the functions Exp (iωt) and Exp (−iωt) are not vanished. Because these functions are also the solutions of the homogeneous equation associated to Eq.(30). Again to to avoid the appearance of these secular terms in the solutions of Eq.(30) with respect to the fast variable t, the coefficients s2 (τ, σ) and s¯2 (τ, σ) have to take zero values.
here
γ(t, τ, σ) = ωt − (72α2 (σ)/ω)τ + f (σ).
∂ Z¯1 (τ, σ) ∂τ
Using the initial condition in Eq.(19), the general first approximated solution (z0 ) is given by z0 (t, τ, σ) = 2α(σ) cos γ(t, τ, σ), where α(0) = 7
1 Z , f (0) = nπ , n ∈ Z 2
(33)
while the general second approximated solution (z1 ) can be written in the following form
7. Numerical results
In this section a comparison will be investigated among the numerical, first and second approximated solutions of the Sitnikov problem. The investigations include the (34) 12α3 (σ) cos 3γ(t, τ, σ), − numerical solutions of Eq.(4), the first and second apω2 proximated solutions of Eq.(8) via multiple scales method, and the initial conditions are given in Eqs.(29) where γ(t, τ, σ) which are given in Eq.(37) and Eq.(38) respectively. is also given by Eq.(27). We construct the comparison within frame a three different initial conditions. In general the infinitesimal body The solutions in Eq.(33) and Eq.(34) include the three will start its motion with zero velocity (z(0) ˙ = 0), for arbitrary functions Z1 (τ, σ), α(σ) and f (σ) in slow varia three different positions (z(0) = 0.4, 0.8, 1.2), therefor ables τ and σ. But these two solutions must be conve(α0 = 0.2, 0.4, 0.6). The investigation for the motion of nient with the initial conditions in Eq.(29). It is clear that the infinitesimal body will be constructed in two main the only restriction on the function f (σ) is f (0) = 2nπ groups. In the first main groups a three versions for the and there is no restriction on α(σ), but for simplicity we same solution will be showed, for a three different initial will select these two functions as constants, where α(σ) = conditions. Then we mean that the changes with respect Z/2 = α0 and f (σ) = 2nπ, thereby we will impose that to the initial conditions, not the kind of obtained solution. f (0) = 2nπ instead of f (0) = nπ in order to avoid a conThis group will include Figs.(5, 6, 7), in this figures the tradiction between the obtained results from the condition red, green and blue curves refer to α0 = 0.2, α0 = 0.4 and in Eq.(19) and the other from the condition in Eq.(29). To α0 = 0.6 respectively. While in the Second main groups a make a coherent convenient or a harmony among the sothree different solutions will be showed, for the same inilutions and the initial conditions, we have to select also tial condition. Therefore we mean that the changes with Z1 (τ, σ) = Z¯1 (τ, σ) = (6α03 /ω 2 ) cos 3f0 . Thereby the first respect to the kind of obtained solution. This group will two solutions in the perturbation series are controlled by include Figs.(8, 9, 10), but in these figures the red, green z0 (t, τ ) = 2α0 cos ψ(t, τ ), and blue curves indicate to the numerical solution (NS), 3 (35) first approximated solution (FA) and second approximated 12α0 [cos ωt − cos 3 ψ(t, τ )] , z1 (t, τ ) = solution (SA) of multiple scales method respectively. 2 ω In Fig. 5, a comparison among three versions for the where numerical solution are showed when the infinitesimal body 72α02 τ + 2nπ. ψ(t, τ ) = ωt − starts its motion from three different positions with respect ω to the common center of primaries mass. It is clear that the motion is periodic and its amplitude will decrease when Since cos(θ+2nπ) = cos θ, then the solutions in Eqs.(35) the infinitesimal body starts its motion from a nearer posican be reduced to a simple forms without losing the gention due to the common center of primaries mass, and vice erality to versa. More reading the initial conditions play a vital role in the numerical solution behaviour, and may the obtained 2 72α z0 (t, τ ) = 2α0 cos ω t − 2 0 τ , motion by the numerical solution is not periodic, when the ω infinitesimal body starts its motion from a point is very far (36) 12α03 72α02 from the common center of primaries mass. But the soluz1 (t, τ ) = cos ωt − cos 3 ω t − 2 τ . ω2 ω tion may be periodic and the motion has regularity in its periodicity for an appropriate choice of initial conditions. The slow variable τ = εt, but with the method of place In Fig. 6, a comparison among three versions for the keeping parameters, the parameter ε will be replace by first approximated solutions (FA) when the infinitesimal one. Using Eqs.(36) then the first approximated solution starts its motion under the same conditions of the numerof multiple scales method is given by ical solution. There is a convenient between the obtained 72α02 motion by the first approximated and numerical solutions (1) t, (37) z (t) = 2α0 cos ω 1 − 2 ω in periodic behaviour, however the periods number of motion are different in both cases, when the infinitesimal while the second approximated solution of multiple scales body starts its motion from a point far from the common method is given by center of primaries mass and vice versa. 2 72α The behaviour of motion within frame of the second z (2) (t) = 2α0 cos ω 1 − 2 0 t approximated solution (SA) is similar to the behaviour ω (38) of motion within frame of the first approximated solution 3 2 12α 72α + 2 0 cos ωt − cos 3 ω 1 − 2 0 t . (FA), when the infinitesimal body starts its motion from ω ω z1 (t, τ, σ) = Z1 (τ, σ)eiωt + Z¯1 (τ, σ)e−iωt
8
0 =0.2
0 =0.4
0 =0.6
0 =0.2
0 =0.4
0 =0.6
1.5 1.0 1.0
0.5 z(t)
z(t)
0.5
0.0
0.0
-0.5
-0.5
-1.0 -1.0 -1.5 0
5
10
15
20
0
5
10
t
Figure 5: Numerical solution for three different values of initial conditions (Color figure online).
0 =0.2
0 =0.4
15
20
t
Figure 7: Second approximated solution for three different values of initial conditions (Color figure online).
0 =0.6
dent within frame of the first and second approximated solutions, when the infinitesimal body starts its motion from a nearer point due to the common center of primaries mass.
1.0
0.5
z(t)
NS
FA
SA
0.0 0.4 -0.5 0.2
0
5
10
15
z(t)
-1.0
20
0.0
t
Figure 6: First approximated solution for three different values of initial conditions (Color figure online).
-0.2
a nearer position due to the common center of primaries mass, the motion will be regular and periodic. The chaotic behaviour may appear in the obtained motion by the second approximated solution when the infinitesimal body starts its motion from a point far from the common center of primaries mass, see Fig. 7. In general the obtained motion by the multiple scales method will be periodic all time.
-0.4 0
5
10
15
20
t
Figure 8: Comparison among numerical, first and second approximated solutions at α0 = 0.2 (Color figure online).
NS
In Figs. (8, 9, 10) comparisons among the numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) are showed within frame the same initial conditions. It is clear that the motion are periodic due to the numerical solutions. But the periods number of motion is inversely proportional with the start point distance from the common center of primaries mass, unlike for the obtained motion by the first and second approximated solutions. On the other hand the amplitude of motion in the three solutions is directly proportional with the separation distance between the start point motion and the common center of primaries mass. Furthermore the behaviour of motion is the same and may be coinci-
FA
SA
z(t)
0.5
0.0
-0.5
0
5
10
15
20
t
Figure 9: Comparison among numerical, first and second approximated solutions at α0 = 0.4 (Color figure online).
9
NS
FA
SA
comparisons have been developed among three different solutions at the same initial conditions.
1.0
Furthermore we demonstrate that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The provided motion by the numerical solution is periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. While the motion may not be periodic when the infinitesimal body starts its motion from a point which is very far from the common center of primaries mass.
z(t)
0.5
0.0
-0.5
Regard to the first approximated solution by multiple scale solutions, the motion is periodic all time, with a regularity in its periodicity, whatever the initial conditions. This property is also satisfied for the second approximated solution when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. While the motion may be a chaotic when the infinitesimal body starts its motion from a point is far from the common center of primaries mass, however it is also periodic motion all time .
-1.0
0
5
10
15
20
t
Figure 10: Comparison among numerical, first and second approximated solutions at α0 = 0.6 (Color figure online).
More reading for the Figs. (5–10) the motion is periodic and the changes may be in the difference of its amplitude, that is considered the major difference between the obtained motion by the numerical solution and the other by the multiple scales solutions. It is also observed that when the infinitesimal body starts its motion from a point far from the common center of primaries mass, the obtained motion by the numerical may not be periodic, while a chaotic behaviour will appear in the obtained motion by the second approximated solution. The chaotic behaviour in motion is considered the major difference between the obtained motion by the first and second approximated solutions via multiple scales method. Regard to the multiple scale solutions, the motion is periodic all time, with regularity in its periodicity when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. While the motion may be chaotic when the infinitesimal body starts its motion from a point is far from the common center of primaries mass due to the second approximated solution, however it is periodic motion.
Now we summarize the status of motion regard to each solution in the following items: • The obtained motion by the first approximated solution is regular periodic motion all time whatever initial conditions. • The obtained motion by the numerical and the second approximated solutions is periodic when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. • When the infinitesimal body starts its motion from a point which is very far from the common center of primaries mass, the obtained motion by the numerical solution may not be periodic, while the obtained motion by the second approximated solution may be a chaotic, however it is also periodic all time. • The major difference between the obtained periodic motion by the numerical solution and the others by the approximated solutions is in the difference of its amplitude.
8. Conclusion In this paper two approximated analytical periodic solutions of Sitnikov problem have been developed. The multiple scales method is used to remove the secular terms and find periodic solutions in closed forms. At the same time of removing the secular terms the periodicity conditions for the approximated solutions are constructed.
• The major difference between the first and second approximated solutions of multiple scales method is a chaotic motion behaviour. Which will appear due to the provided solution by the second approximated solution when the infinitesimal body starts its motion from a point which is far from the common center of primaries mass.
Comparisons among numerical solutions (NS), the first approximated solutions (FA) and the second approximated solutions (SA) are investigated graphically in two main groups. In the first group a comparison among three issues for the same solution is developed by depending on a three different initial conditions. The different of these conditions means that the infinitesimal body starts its motion from three different positions from the common center of primaries mass. While in the second main group the
Finally we underline that the obtained solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the higher order approximation has more accuracy than the lower approximation. Moreover the solutions of multiple scales tech10
niques are more realistic than the numerical solution, because the numerical solution may not represent a periodic motion for a long time.
[18] E. I. Abouelmagd, A. A. Ansari, The motion properties of the infinitesimal body in the framework of bicircular sun perturbed earth-moon system. [19] G. Pavanini, Sopra una nuova categoria di soluzioni periodiche nel problema dei tre corpi, Annali di Matematica Pura ed Applicata (1898-1922) 13 (1) (1907) 179–202. [20] W. D. MacMillan, An integrable case in the restricted problem of three bodies, The Astronomical Journal 27 (1911) 11–13. [21] K. Sitnikov, The existence of oscillatory motions in the threebody problem, in: Soviet Physics Doklady, Vol. 5, 1961, p. 647. [22] S. B. Faruque, Solution of the Sitnikov problem, Celestial Mechanics and Dynamical Astronomy 87 (4) (2003) 353–369. [23] M. d. S. Suraj, M. R. Hassan, Sitnikov restricted four-body problem with radiation pressure, Astrophysics and Space Science 349 (2) (2014) 705–716. [24] R. Ortega, Symmetric periodic solutions in the Sitnikov problem, Archiv der Mathematik 107 (4) (2016) 405–412. [25] E. E. Zotos, M. S. Suraj, R. Aggarwal, S. K. Satya, Investigating the basins of convergence in the circular sitnikov three-body problem with non-spherical primaries, Few-Body Systems 59 (4) (2018) 69. [26] T. Bountis, K. E. Papadakis, The stability of vertical motion in the N-body circular Sitnikov problem, Celestial Mechanics and Dynamical Astronomy 104 (1-2) (2009) 205–225. [27] V. V. Sidorenko, On the circular Sitnikov problem: the alternation of stability and instability in the family of vertical motions, Celestial Mechanics and Dynamical Astronomy 109 (4) (2011) 367–384. [28] J. Llibre, R. Ortega, On the families of periodic orbits of the Sitnikov problem, SIAM Journal on Applied Dynamical Systems 7 (2) (2008) 561–576. [29] V. S. Kalantonis, E. A. Perdios, A. E. Perdiou, The Sitnikov family and the associated families of 3D periodic orbits in the photogravitational RTBP with oblateness, Astrophysics and Space Science 315 (1-4) (2008) 323–334. [30] C. D. Karlgaard, F. H. Lutze, Second-order relative motion equations, Journal of Guidance, Control, and Dynamics 26 (1) (2003) 41–49.
Acknowledgements The first and second authors are partially supported by Fundaci´on S´eneca (Spain), grant 20783/PI/18, and Ministry of Science, Innovation and Universities (Spain), grant PGC2018-097198-B-100 References [1] Z. E. Musielak, B. Quarles, The three-body problem, Reports on Progress in Physics 77 (6) (2014) 065901. [2] K. Meyer, D. Schmidt, From the restricted to the full three– body problem, Transactions of the American Mathematical Society 352 (5) (2000) 2283–2299. ˇ [3] M. Suvakov, V. Dmitraˇsinovi´ c, Three classes of newtonian three-body planar periodic orbits, Physical review letters 110 (11) (2013) 114301. [4] X. Li, S. Liao, More than six hundred new families of newtonian periodic planar collisionless three-body orbits, SCIENCE CHINA Physics, Mechanics & Astronomy 60 (12) (2017) 129511. [5] X. Li, Y. Jing, S. Liao, Over a thousand new periodic orbits of a planar three-body system with unequal masses, Publications of the Astronomical Society of Japan 70 (4) (2018) 64. [6] X. Li, S. Liao, Collisionless periodic orbits in the free-fall threebody problem, New Astronomy 70 (2019) 22–26. [7] T. Luo, M. Xu, Dynamics of the spatial restricted three-body problem stabilized by Hamiltonian structure-preserving control, Nonlinear Dynamics (2018) 1–17. [8] E. I. Abouelmagd, M. A. Sharaf, The motion around the libration points in the restricted three-body problem with the effect of radiation and oblateness, Astrophysics and Space Science 344 (2) (2013) 321–332. [9] E. I. Abouelmagd, M. S. Alhothuali, J. L. G. Guirao, H. M. Malaikah, The effect of zonal harmonic coefficients in the framework of the restricted three-body problem, Advances in Space Research 55 (6) (2015) 1660–1672. [10] S. M. Elshaboury, E. I. Abouelmagd, V. S. Kalantonis, E. A. Perdios, The planar restricted three-body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic orbits, Astrophysics and Space Science 361 (9) (2016) 315. [11] N. Pathak, S. Elshaboury, On the triangular points within frame of the restricted three–body problem when both primaries are triaxial rigid bodies, Applied Mathematics and Nonlinear Sciences 2 (2) (2017) 495–508. [12] E. I. Abouelmagd, F. Alzahrani, J. L. G. Guirao, A. Hobiny, Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl.(JNSA) 9 (4) (2016) 1716–1727. [13] F. Alzahrani, E. I. Abouelmagd, J. L. G. Guirao, A. Hobiny, On the libration collinear points in the restricted three–body problem, Open Physics 15 (1) (2017) 58–67. [14] A. Hanslmeier, R. Dvorak, Numerical integration with lie series, Astronomy and Astrophysics 132 (1984) 203. [15] E. I. Abouelmagd, J. L. G. Guirao, A. Mostafa, Numerical integration of the restricted three-body problem with Lie series, Astrophysics and Space Science 354 (2) (2014) 369–378. [16] E. I. Abouelmagd, A. Mostafa, Out of plane equilibrium points locations and the forbidden movement regions in the restricted three-body problem with variable mass, Astrophysics and Space Science 357 (1) (2015) 58 (10 Page). [17] M. d. S. Suraj, E. I. Abouelmagd, R. Aggarwal, A. Mittal, The analysis of restricted five–body problem within frame of variable mass, New Astronomy 70 (2019) 12–21.
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Periodic solution of the nonlinear Sitnikov restricted three-body problem Elbaz I. Abouelmagd, Juan Luis Garc´ıa Guirao, Ashok Kumar Pal PII: DOI: Reference:
S1384-1076(19)30223-4 https://doi.org/10.1016/j.newast.2019.101319 NEASPA 101319
To appear in:
New Astronomy
Received date: Revised date: Accepted date:
20 July 2019 19 August 2019 15 September 2019
Please cite this article as: Elbaz I. Abouelmagd, Juan Luis Garc´ıa Guirao, Ashok Kumar Pal, Periodic solution of the nonlinear Sitnikov restricted three-body problem, New Astronomy (2019), doi: https://doi.org/10.1016/j.newast.2019.101319
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Highlights • The properties of Sitnikov motion are analyzed. • The multiple scales method is used to remove secular terms. • The periodic solution of the circular Sitnikov problem is found. • Comparisons among a numerical solution and the solutions provide by the multiple scales method are investigated.
1
Periodic solution of the nonlinear Sitnikov restricted three-body problem Elbaz I. Abouelmagda,b , Juan Luis Garc´ıa Guiraoc , Ashok Kumar Pald a Celestial Mechanics and Space Dynamics Research Group (CMSDRG), Astronomy Department, National Research Institute of Astronomy and Geophysics (NRIAG), Helwan – 11421, Cairo, Egypt. b Nonlinear Analysis and Applied Mathematics Research Group (NAAM), Mathematics Department, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia c Departamento de Matem´ atica Aplicada y Estad´ıstica. Universidad Polit´ ecnica de Cartagena, Hospital de Marina, 30203-Cartagena, Regi´ on de Murcia, Spain d Department of Mathematics and Statistics, School of Basic Science, Manipal University Jaipur, Jaipur – 303007, Rajasthan, India
Abstract The objective of this paper is to find periodic solutions of the circular Sitnikov problem by the multiple scales method which is used to remove the secular terms and find the periodic approximated solutions in closed forms. Comparisons among a numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) via multiple scales method are investigated graphically under different initial conditions. We observe that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The obtained motion is periodic, but the difference of its amplitude is directly proportional with the initial conditions. We prove that the obtained motion by the numerical or the second approximated solutions is a regular and periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries. Otherwise when the start point distance of motion is far from this center, the numerical solution may not be represent a periodic motion for along time, while the second approximated solution may present a chaotic motion, however it is always periodic all time. But the obtained motion by the first approximated solution is periodic and has regularity in its periodicity all time. Finally we remark that the provided solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the second approximation has more accuracy than the first approximation. Moreover the solutions of multiple scales technique are more realistic than the numerical solution because there is always a warranty that the motion is periodic all time. Keywords: Sitnikov restricted three–body problem; Multiple scales method; Periodic solution. 1. Introduction One of the most important dynamical systems in celestial mechanics and space mechanics is the three–body problem. It is according to variation of its versions and applications in either stellar dynamics or astrodynamics. The general model of this problem is the three–body problem in which three bodies move in three dimensions under their mutual gravitational interactions, within frame Newton’s laws of motion and universal gravitation, without imposed constraints on the initial positions and velocities vectors of these bodies. An extended and a comprehensive review on the three–body problem is stated by [1, 2]. Although the general three–body problem is common in the field of celestial mechanics, there is no big significant contributions in its study at the literature on the properties of this system and in particular for the periodic Email addresses: [email protected] or [email protected] (Elbaz I. Abouelmagd), [email protected] (Juan Luis Garc´ıa Guirao), [email protected]. (Ashok Kumar Pal) Preprint submitted to New Astronomy
solutions. Note that there were only three families of periodic orbits which have been found until 2013 within frame of zero angular momentum and equal masses, since this problem “(general three–body problem)” was recognized [3]. But in the recent years with the serious work of Xiaoming Li and his team, new results and many families of Newtonian periodic orbits related to three–body problem were found, see for instance [4–6]. We underline that 695 families of Newtonian periodic collision less orbits within frame of equal masses in the three bodies system, with zero angular momentum were found numerically in [4]. Also in the case of two masses equal and the third one different, 1349 new families of periodic orbits have been constructed in [5]. None of these new families were found before, except the famous Figure– Eight family. In other words, 1223 among of 1349 families are entirely new. Furthermore within of frame three–body problem in [6], using a clean numerical simulation strategy for chaotic dynamic systems, 316 collision less periodic orbits are obtained with a few chosen values for the parameter of mass ratio. Authors remarks that 313 collision less September 17, 2019
free-fall periodic orbits obtained are new.
tion of the infinitesimal body are developed in dimensionless variable, such that depends on time t and the eccentricity parameter e of the primaries. The problem is considered a generalization to a “MacMillan problem” which is a special case from the elliptical Sitnikov problem when e = 0. In the case of e 6= 0 the Sitnikov model represents a chaotic system and admits many different kinds of motions, whoever it has a simple form than the ones found in chaotic dynamical systems. One of the most important properties of this problem is its mathematical simple structure, but at the same time it has a complexity on its possible type of motion. These properties make the Sitnikov model a unique system the field of stellar dynamics.
Within framework of imposed restrictions on the third body, where it has negligible mass with respect to the other two bodies and moving under the gravitational effects of these two bodies, the three–body problem will be reduced to the restricted problem, in this case the third body is called the infinitesimal body and the others bodies are called the primaries. With more restrictions, if the primaries motion is in the same plane with circular or elliptical orbits, we will get the spatial circular or elliptical restricted three–body problem in which the third body moves in three dimensional space [7]. But in the case of the infinitesimal body and the primaries move in the same plane, we will obtain the most familiar dynamical system in celestial mechanics, which is called the “planar restricted three–body problem”.
[26] studied on the stability and bifurcation of vertical motion into three-dimensional periodic orbits families in the Sitnikov problem within frame of restricted N–body model. [27] studied the circular Sitnikov model by alternation of stability and instability within frame of the family of periodic vertical motions when the varying continuous of monotonic amplitude is considered. [22] computed the solution by the Lindstedt–Poincar´e method and compared with existing solutions. [28] have computed analytically the families of symmetric periodic orbits in the elliptic Sitnikov problem. They also provide a qualitative information on the bifurcation diagram of such families of periodic orbits. [29] studied the photogravitational Sitnikov problem in the restricted three–body problem with oblateness. [24] provided the existence criteria of odd periodic solutions with a prescribed number of zeros.
Within frame of planar circular model, an analytical investigations on the effects of some perturbed forces such that non–sphericity of the primaries in the locations of equilibrium points, and their linear stability, as well as periodic orbits around these points have been developed by [8–11]. Some works also are devoted to investigate the locations of collinear points and the motion in the vicinity of such points, see for instance [12, 13]. The numerical integration of this problem using Lie series is also analyzed in these works [14, 15]. An important work for extending the planar circular restricted three–body to more realistic and effective models in the field of celestial mechanics are addressed by [16–18].
This paper includes eight sections, it is organized in the following way: A brief overview is stated on literature review due to periodic solutions of general three–body problem and Sitnikov problem in Section 1. In Section 2 the model description of the three–body problem to Sitnikov model is addressed. In Section 3, the dynamics characteristics of motion are investigated. In Section 4, the general structure of multiple scales method is stated. The first and second approximated solutions via multiple scales method and the periodicity conditions are constructed and explained in Sections 5 and 6. In addition the numerical results and a comparisons among the obtained solutions are investigated in Section 7. Finally the conclusion are stated in Section 8.
One of the reduced models of the three–body problem is the Sitnikov problem, which is considered a sub–case from the spatial elliptic restricted three–body problem, where the infinitesimal body (third body) has oscillation motion along Z−axis, which is perpendicular to the orbital plane of the primaries. In this model the primaries have equal masses and rotate in elliptical orbits. This model is called elliptical Sitnikov problem, that was constructed by [19]. The case when the primaries move in circular orbits around their common center of mass, is called the circular Sitnikov problem, which was stated by [20]. The Sitnikov models are considered the simplest sub– cases of general N −−body problem which can be used as a first approximation in many cases of real situations for astronomical problems. The existence of oscillation motion for this problem is proved for first time by [21], thereby it is called Sitnikov problem. It has also been studied by several researcher such as [22–24]. The Newton–Raphson basins of convergence for the equilibrium points in this problem are also explored numerically by [25]. These basins are studied under the sphericity lack of primaries. They also revealed the attracting regions with several types of two dimensional planes, by using the classical Newton–Raphson iterative method.
2. Model description Let m1 , m2 and m the three masses of the bodies b1 , b2 and b3 where b1 and b2 are the primaries, while b is the third body. One of the versions of the three–body problem is the spatial circular restricted three–body problem. In which the primaries move in the same plane in circular orbits around their barycenter with the normalized angular velocity to its motion plane. The motion of third body, which called the infinitesimal body is free in order to move in the 3−dimensional space, and its motion does not affect
In the elliptical Sitnikov problem, the equation of mo3
the primaries motion. We also assume that the total mass of the primaries is one, in which m1 = 1 − µ and m2 = µ.
Eqs.(1, 2, 3) can be reduced to the circular Sitnikov problem or the motion of the test particle along the vertical Z−axis d V (z), (4) z¨ = dz where 1 V (z) = p . (5) 1/4 + z 2
By choosing a synodic coordinates system with the origin at the center of mass, as well as the primaries b1 and b2 are fixed on the X−axis at (−µ, 0, 0) and (1 − µ, 0, 0) respectively. Now we assume that r, r1 and r2 are the positions vectors of the infinitesimal body with respect to the origin of the synodic coordinates and the primaries. Furthermore the scale of the time is chosen, so that the gravitational constant is one. Then the equations of motion of the infinitesimal body will be controlled by x ¨ − 2y˙ = Wx , y¨ + 2x˙ = Wy ,
3. Properties of Sitnikov motion The motion of a test particle in Sitnikov model is governed by a nonlinear differential equation as in Eq.(4), where the only exact analytical solution can be found in terms of elliptic functions, which is not easy in its mathematical structures. Thereby the alternative possible solutions with mathematical forms for this equation are an approximated solutions, which reflect the original features of the exact solution. These solutions can be developed by using the perturbation techniques. Although the perturbation techniques are considered effective tools in obtaining a solution of nonlinear differential equations, but these techniques have their own limitations. Because they depend on a very small physical parameter, which does not appear in Eq.(4). In fact the most applications of perturbation methods are not applied without the existence of this parameter.
(1)
z¨ = Wz , where W (x, y, z) = and r= r1 = r2 = here
p p
p
µ 1−µ 1 2 (x + y 2 ) + + , 2 r1 r2
(2)
x2 + y 2 + z 2 , (x + µ)2 + y 2 + z 2 ,
(3)
(x + µ − 1)2 + y 2 + z 2 ,
r = |r| , r1 = |r1 | , and r2 = |r2 |.
It is clear that from the Fig.2, the potential is symmetric around z = 0, which means that the potential may represent a harmonic oscillator motion.
The spatial restricted three–body problem is not only a special model from three–body problem, but also the Sitnikov problem, which can be considered it as a special model from the general problem or from the spatial restricted problem, when the primaries are equal in mass and the infinitesimal body moves only onpZ−axis. Hence x = y = 0 , µ = 1/2 and r1 = r2 = 1/4 + z 2 . See Fig.1 for the configuration of this model. In this context,
V(z)
-0.5
-1.0
-1.5
-2.0 -4
-2
0
2
4
z Figure 2: The curve of the potential V (z) (Color figure online).
Let E be the total mechanical energy, then 1 2 1 z˙ − p = E. 2 1/4 + z 2
(6)
The region of motion can be investigated with a simple analysis of Eq.(6). Since the potential value V (z) ∈ [−2, 0) for all values of z. Hence the motion will undefined in the rang E < −2 and bounded for −2 < E < 0, while
Figure 1: The configuration of Sitnikov problem (Color figure online).
4
it will be unbounded in the rang E > 0. The region of motion in 3–dimensional and its phase portrait are drawn in Fig. 3 and Fig. 4. In phase portrait figure, the regions
bounded and the third body will also enforced to remain at the center of masses. This is clear from the phase portrait in Fig. 3 when the value of total energy (E) is tends to (−2). Within frame of the total energy is negative and it is close to −2, then the third body will be near and close to the center of masses, thereby |z| 1. Then with a help of Eqs.(4, 5) the equation of motion up O(z 3 ) can be written as z¨ + ω 2 z − $2 z 3 = 0, (7) where ω 2 = 8 and $2 = 48. To find the periodic solution of Eq.(7), an approximation may still be found by introducing a very small parameter into the problem, but in the end its value will replace by one. This method is acceptable when the terms recalling the differential equation are quasi-linear or small in of themselves. Parameters introduced in this way are always called “Place Keeping Parameters“, see for more details [30]. Now we interchange each z by δz where δ 1, and take ε = δ 2 , of course, also ε 1, then Eq.(7) will take the following form
Figure 3: The region of possible motion in the 3–dimensional when E < 0 (Color figure online).
2
z¨ + ω 2 z − ε$2 z 3 = 0.
0.5
(8)
Then Eq.(8) represents equation of motion in standard form to apply one of the perturbation methods, like multiple scales method or Lindstedt–Poincar´e technique.
1 0
dz/dt
4. Multiple scales solution 0
Let the initial conditions of motion be z(0) = Z, z(0) ˙ = 0.
-1
(9)
0
0
To continue in our procedures, we will write the solution z(t) in the form of a perturbation series: 1
z(t) =
-2 -2
-1
0
1
2
∞ X
εn zn (t, τ, σ),
(10)
n=0
z Figure 4: The phase portrait of motion for different values of total energy E (Color figure online).
where τ = εt and σ = ε2 t.
of periodic motion are determined by closed contours when −2 < E < 0, and the region of unbounded motion will be determined when E ≥ 0.
In fact, the provided solution by multiple scales method includes the fast and slow variables. Although these variables are dependent, the processing strategy of multiple scales method for removing the secular terms is handling the two slow variables as independent variables. Hence the ordinary derivative with respect to the fast variable t will be expanded to the differential operator Dt . Which includes the partial derivatives with respect to the fast variable t and the slow variables τ and σ. With the help of the chain rule, the differential operator Dt is defined as
Furthermore the third body will has zero velocity (z˙ = 0) in three cases. The first two cases when it tends to infinity (z → ±∞) with zero energy (E = 0), then there are two stationary points at positive (negative) infinity. In these two case the motion is considered unbounded and the third body will be enforced to remain stationary at infinity. While the third case when total energy tends to −2 (E = −2) and the infinitesimal body moves to stay at the common center of masses (z = 0), then the motion is
∂ ∂ ∂ d = Dt := ( + ε + ε2 ), dt ∂t ∂τ ∂σ 5
(11)
hence
∞ X d z(t) = Dt εn zn (t, τ, σ). dt n=0
with the initial conditions Z0 (0, 0) + Z¯0 (0, 0) = Z,
(12)
With the conditions that zn (t, τ, σ) is continuous and differentiable with respect to t, τ and σ, and substituting Eq.(11) into Eq.(12) with some simple calculations, one obtains ∂z0 ∂z1 ∂z0 dz = +ε + dt ∂t ∂t ∂τ (13) ∂z2 ∂z1 ∂z0 2 3 +ε + + + O(ε ), ∂t ∂τ ∂σ 2 ∂ 2 z0 ∂ z1 ∂ 2 z0 d2 z = + ε + 2 dt2 ∂t2 ∂t2 ∂τ ∂t 2 ∂ z2 ∂ 2 z1 (14) +2 ∂t2 ∂τ ∂t 3 2 + O(ε ). +ε ∂ 2 z0 ∂ 2 z0 + +2 ∂σ∂t ∂τ 2 Using Eqs.(10, 13, 14) and Eq.(8), with the help of Eq.(9) and equalling the coefficients which has the same order in ε, we will get a series of linear partial differential equations which govern the functions of zi (t, τ, σ), for i = 0, 1, 2, these equations are ∂ 2 z0 = −ω 2 z0 , ∂t2 z0 (0, 0, 0) = Z,
∂ z0 (0, 0, 0) = 0. ∂t
∂ 2 z1 ∂ 2 z0 2 3 = −ω z + 48z − 2 , 1 0 ∂t2 ∂τ ∂t z1 (0, 0, 0) = 0, ∂ ∂ z1 (0, 0, 0) = − z0 (0, 0, 0). ∂t ∂τ
Here Z0 (τ, σ) is an arbitrary complex function in the slow variable τ , σ and Z¯0 (τ, σ) is its complex conjugate. 5. First approximated solution Insert Eq.(18) into Eq.(16), one obtains ∂ 2 z1 = − w2 z1 ∂t2 + 48 Z03 (τ, σ)ei3ωt + Z¯03 (τ, σ)e−i3ωt
(20)
+ s1 (τ, σ)eiωt + s¯1 (τ, σ)e−iωt ,
where
∂Z0 s1 (τ, σ) = 2 72Z02 Z¯0 − iω , ∂τ ∂ Z¯0 . s¯1 (τ, σ) = 2 72Z¯02 Z0 + iω ∂τ
(21) (22)
It is clear that the solution of Eq.(20) will contain a secular term, if the coefficients of the functions Exp (iωt) and Exp (−iωt) have non zero values. Because these functions are the solutions of the homogeneous equation associated to Eq.(20). In order to avoid the appearance of these secular terms in the solutions of Eq.(20) with respect to the fast variable t, the coefficients s1 (τ, σ) and s¯1 (τ, σ) have to equal zero.
(15)
5.1. Periodicity condition of first approximation Equalling coefficients s1 (τ, σ) and s¯1 (τ, σ) to zero not only to avoid the secular terms but also to determine the function Z0 (τ, σ) and Z¯0 (τ, σ) with an elegant way, but s1 (τ, σ) and s¯1 (τ, σ) are just two conjugate quantities. It is enough to solve Eq.(21) with s1 (τ, σ) = 0, because the solution of Eq.(22) for s¯1 (τ, σ) = 0 will give the same results for the solution of Eq.(21). This processing will give the warranty that the solution of z1 (t, τ, σ) will not involve secular terms.
(16)
∂ 2 z2 = −ω 2 z2 + 144z02 z1 ∂t2 ∂ 2 z1 ∂ 2 z0 ∂ 2 z0 −2 −2 − , (17) ∂τ ∂t ∂σ∂t ∂τ 2 z2 (0, 0, 0) = 0, ∂ ∂ ∂ z2 (0, 0, 0) = − z1 (0, 0, 0) − z0 (0, 0, 0). ∂t ∂τ ∂σ
To carry out our purpose, we will solve Eq.(21) with the help of the polar coordinates (R, θ). Hence we suppose that Z0 (τ, σ) = R(τ, σ)eiθ(τ,σ) ,
To accomplish our steps, we have to find the solutions of Eqs.(15–17) without including secular terms, specifically, the solutions are periodic functions in the fast variable t. This means that we have to determine the formulas of the functions zi (t, τ, σ) where zi (t + 2π, τ, σ) = zi (t, τ, σ). Now we will assume that the solution of Eq.(15) in the independently handled variables t, τ and σ is z0 (t, τ, σ) = Z0 (τ, σ)eiωt + Z¯0 (τ, σ)e−iωt ,
(19)
Z0 (0, 0) − Z¯0 (0, 0)) = 0.
(23)
R, θ : R → R,
insert Eq.(23) into Eq.(21), and use the condition of s1 (τ, σ) = 0, one obtains
Z0 (τ, σ) = α(σ)e
(18) Z¯0 (τ, σ) = α(σ)e 6
if (σ)−
72α2 (σ) τ ω
−if (σ)−
2
,
72α (σ) τ ω
(24) ,
s¯2 (τ, σ) = 288α2 (σ)Z¯1 (τ, σ) + 2iω
where α(σ) and f (σ) are arbitrary functions in the slow variable σ, with
+ 144α2 (σ)Z1 (τ, σ)e−2iθ(τ,σ) dα(σ) iω 2 ω + 144iτ α2 (σ) 2e−iθ(τ,σ) dσ + ω2 df (σ) +3024α5 (σ) + ω 3 α(σ) dσ
R(τ, σ) = α(σ), (25)
2
72α (σ) θ(τ, σ) = f (σ) − τ. ω
Substituting Eqs. (24),(25) into Eq.(18), then we get the first approximated periodic solution in form i h (26) z0 (t, τ, σ) = α(σ) eiγ(t,τ,σ) + e−iγ(t,τ,σ) , (27)
Now Eq.(20) can be written as ∂ 2 z1 = −w2 z1 Z03 (τ, σ)ei3ωt + Z¯03 (τ, σ)e−i3ωt , ∂t2 and its general solution is
Z1 (0, 0) + Z¯1 (0, 0) =
here i=
√
(29)
6. Second approximated solution Substituting Eq.(26) and Eq.(28) into Eq.(17), then the solution of z2 (t, τ, σ) is controlled by ∂ 2 z2 = − w2 z2 ∂t2 864α5 (σ) i5γ(t,τ,σ) −i5γ(t,τ,σ) + e + e ω2 6048α5 (σ) i3γ(t,τ,σ) −i3γ(t,τ,σ) (30) e + e − ω2 " # Z1 (τ, σ)ei(ωt+2θ(τ,σ)) + + 144α2 (σ) Z¯1 (τ, σ)e−i(ωt+2θ(τ,σ))
Let us go back to periodicity condition s2 (τ, σ) = 0, which leads to a system of partial differential equations, and there is extra difficulty to find its analytical solution in most cases. This is considered the major defect of multiple scales method. In order to avoid this difficulty, we have to note that we search for an approximated analytical solution of the second approximated (z1 ), we mean that the solution does not general or exact solution. Hence the particular solution of proper selection of the arbitrary functions which make a convenient with the initial conditions may be acceptable, if its validation is tested.
+ s2 (τ, σ)eiωt + s¯2 (τ, σ)e−iωt , where ∂Z1 (τ, σ) ∂τ
+ 144α2 (σ)Z¯1 (τ, σ)e2iθ(τ,σ) (31) 2 dα(σ) 2 iθ(τ,σ) iω ω − 144iτ α (σ) 2e dσ − , 2 df (σ) ω −3024α5 (σ) − ω 3 α(σ) dσ
.
One has to note that we arrange the equation of z1 , within frame its solution has no secular terms or is not resonance with the solution of homogenous part to find the arbitrary functions Z0 (τ, σ) and Z¯0 (τ, σ). The same processing will be done to find the arbitrary functions Z1 (τ, σ) and Z¯1 (τ, σ), within frame the solution of z2 has no secular terms. That will be repeated n times to get the periodic function, which represents the solution of zn .
−1, α0 = α(0) and f0 = f (0).
s2 (τ, σ) = 288α2 (σ)Z1 (τ, σ) − 2iω
(32)
6.1. Periodicity condition of second approximation The periodicity condition of obtaining periodic solution for the second approximation is available through the solution of Eq.(31) with s2 (τ, σ) = 0 and Eq.(32) with s¯2 (τ, σ) = 0. But it is also enough to solve Eq.(31) with s2 (τ, σ) = 0. Because the output solutions of s2 (τ, σ) = 0 and s¯2 (τ, σ) = 0 will be the same. This processing will also give the warranty that the solution of z2 will not include secular or unbounded terms and at least there are no secularities appear in the solution of perturbation series up to the third term. Furthermore we can construct the arbitrary functions Z1 (τ, σ) and Z¯1 (τ, σ).
z1 (t, τ, σ) = Z1 (τ, σ)eiωt + Z¯1 (τ, σ)e−iωt , i (28) 6α3 (σ) h i3γ(t,τ,σ) −i3γ(t,τ,σ) e + e − , ω2 with the initial conditions 12 3 α cos 3f0 , ω2 0 36α03 Z1 (0, 0) − Z¯1 (0, 0) = i [4 sin f0 + sin 3f0 ] , ω2
As a similar way in the first approximated solution, the solution of Eq.(30) will contain secular terms, if the coefficients s2 (τ, σ) and s¯2 (τ, σ) of the functions Exp (iωt) and Exp (−iωt) are not vanished. Because these functions are also the solutions of the homogeneous equation associated to Eq.(30). Again to to avoid the appearance of these secular terms in the solutions of Eq.(30) with respect to the fast variable t, the coefficients s2 (τ, σ) and s¯2 (τ, σ) have to take zero values.
here
γ(t, τ, σ) = ωt − (72α2 (σ)/ω)τ + f (σ).
∂ Z¯1 (τ, σ) ∂τ
Using the initial condition in Eq.(19), the general first approximated solution (z0 ) is given by z0 (t, τ, σ) = 2α(σ) cos γ(t, τ, σ), where α(0) = 7
1 Z , f (0) = nπ , n ∈ Z 2
(33)
while the general second approximated solution (z1 ) can be written in the following form
7. Numerical results
In this section a comparison will be investigated among the numerical, first and second approximated solutions of the Sitnikov problem. The investigations include the (34) 12α3 (σ) cos 3γ(t, τ, σ), − numerical solutions of Eq.(4), the first and second apω2 proximated solutions of Eq.(8) via multiple scales method, and the initial conditions are given in Eqs.(29) where γ(t, τ, σ) which are given in Eq.(37) and Eq.(38) respectively. is also given by Eq.(27). We construct the comparison within frame a three different initial conditions. In general the infinitesimal body The solutions in Eq.(33) and Eq.(34) include the three will start its motion with zero velocity (z(0) ˙ = 0), for arbitrary functions Z1 (τ, σ), α(σ) and f (σ) in slow varia three different positions (z(0) = 0.4, 0.8, 1.2), therefor ables τ and σ. But these two solutions must be conve(α0 = 0.2, 0.4, 0.6). The investigation for the motion of nient with the initial conditions in Eq.(29). It is clear that the infinitesimal body will be constructed in two main the only restriction on the function f (σ) is f (0) = 2nπ groups. In the first main groups a three versions for the and there is no restriction on α(σ), but for simplicity we same solution will be showed, for a three different initial will select these two functions as constants, where α(σ) = conditions. Then we mean that the changes with respect Z/2 = α0 and f (σ) = 2nπ, thereby we will impose that to the initial conditions, not the kind of obtained solution. f (0) = 2nπ instead of f (0) = nπ in order to avoid a conThis group will include Figs.(5, 6, 7), in this figures the tradiction between the obtained results from the condition red, green and blue curves refer to α0 = 0.2, α0 = 0.4 and in Eq.(19) and the other from the condition in Eq.(29). To α0 = 0.6 respectively. While in the Second main groups a make a coherent convenient or a harmony among the sothree different solutions will be showed, for the same inilutions and the initial conditions, we have to select also tial condition. Therefore we mean that the changes with Z1 (τ, σ) = Z¯1 (τ, σ) = (6α03 /ω 2 ) cos 3f0 . Thereby the first respect to the kind of obtained solution. This group will two solutions in the perturbation series are controlled by include Figs.(8, 9, 10), but in these figures the red, green z0 (t, τ ) = 2α0 cos ψ(t, τ ), and blue curves indicate to the numerical solution (NS), 3 (35) first approximated solution (FA) and second approximated 12α0 [cos ωt − cos 3 ψ(t, τ )] , z1 (t, τ ) = solution (SA) of multiple scales method respectively. 2 ω In Fig. 5, a comparison among three versions for the where numerical solution are showed when the infinitesimal body 72α02 τ + 2nπ. ψ(t, τ ) = ωt − starts its motion from three different positions with respect ω to the common center of primaries mass. It is clear that the motion is periodic and its amplitude will decrease when Since cos(θ+2nπ) = cos θ, then the solutions in Eqs.(35) the infinitesimal body starts its motion from a nearer posican be reduced to a simple forms without losing the gention due to the common center of primaries mass, and vice erality to versa. More reading the initial conditions play a vital role in the numerical solution behaviour, and may the obtained 2 72α z0 (t, τ ) = 2α0 cos ω t − 2 0 τ , motion by the numerical solution is not periodic, when the ω infinitesimal body starts its motion from a point is very far (36) 12α03 72α02 from the common center of primaries mass. But the soluz1 (t, τ ) = cos ωt − cos 3 ω t − 2 τ . ω2 ω tion may be periodic and the motion has regularity in its periodicity for an appropriate choice of initial conditions. The slow variable τ = εt, but with the method of place In Fig. 6, a comparison among three versions for the keeping parameters, the parameter ε will be replace by first approximated solutions (FA) when the infinitesimal one. Using Eqs.(36) then the first approximated solution starts its motion under the same conditions of the numerof multiple scales method is given by ical solution. There is a convenient between the obtained 72α02 motion by the first approximated and numerical solutions (1) t, (37) z (t) = 2α0 cos ω 1 − 2 ω in periodic behaviour, however the periods number of motion are different in both cases, when the infinitesimal while the second approximated solution of multiple scales body starts its motion from a point far from the common method is given by center of primaries mass and vice versa. 2 72α The behaviour of motion within frame of the second z (2) (t) = 2α0 cos ω 1 − 2 0 t approximated solution (SA) is similar to the behaviour ω (38) of motion within frame of the first approximated solution 3 2 12α 72α + 2 0 cos ωt − cos 3 ω 1 − 2 0 t . (FA), when the infinitesimal body starts its motion from ω ω z1 (t, τ, σ) = Z1 (τ, σ)eiωt + Z¯1 (τ, σ)e−iωt
8
0 =0.2
0 =0.4
0 =0.6
0 =0.2
0 =0.4
0 =0.6
1.5 1.0 1.0
0.5 z(t)
z(t)
0.5
0.0
0.0
-0.5
-0.5
-1.0 -1.0 -1.5 0
5
10
15
20
0
5
10
t
Figure 5: Numerical solution for three different values of initial conditions (Color figure online).
0 =0.2
0 =0.4
15
20
t
Figure 7: Second approximated solution for three different values of initial conditions (Color figure online).
0 =0.6
dent within frame of the first and second approximated solutions, when the infinitesimal body starts its motion from a nearer point due to the common center of primaries mass.
1.0
0.5
z(t)
NS
FA
SA
0.0 0.4 -0.5 0.2
0
5
10
15
z(t)
-1.0
20
0.0
t
Figure 6: First approximated solution for three different values of initial conditions (Color figure online).
-0.2
a nearer position due to the common center of primaries mass, the motion will be regular and periodic. The chaotic behaviour may appear in the obtained motion by the second approximated solution when the infinitesimal body starts its motion from a point far from the common center of primaries mass, see Fig. 7. In general the obtained motion by the multiple scales method will be periodic all time.
-0.4 0
5
10
15
20
t
Figure 8: Comparison among numerical, first and second approximated solutions at α0 = 0.2 (Color figure online).
NS
In Figs. (8, 9, 10) comparisons among the numerical solution (NS), the first approximated solution (FA) and the second approximated solution (SA) are showed within frame the same initial conditions. It is clear that the motion are periodic due to the numerical solutions. But the periods number of motion is inversely proportional with the start point distance from the common center of primaries mass, unlike for the obtained motion by the first and second approximated solutions. On the other hand the amplitude of motion in the three solutions is directly proportional with the separation distance between the start point motion and the common center of primaries mass. Furthermore the behaviour of motion is the same and may be coinci-
FA
SA
z(t)
0.5
0.0
-0.5
0
5
10
15
20
t
Figure 9: Comparison among numerical, first and second approximated solutions at α0 = 0.4 (Color figure online).
9
NS
FA
SA
comparisons have been developed among three different solutions at the same initial conditions.
1.0
Furthermore we demonstrate that the initial conditions play a vital role in the numerical and approximated solutions behaviour. The provided motion by the numerical solution is periodic, when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. While the motion may not be periodic when the infinitesimal body starts its motion from a point which is very far from the common center of primaries mass.
z(t)
0.5
0.0
-0.5
Regard to the first approximated solution by multiple scale solutions, the motion is periodic all time, with a regularity in its periodicity, whatever the initial conditions. This property is also satisfied for the second approximated solution when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. While the motion may be a chaotic when the infinitesimal body starts its motion from a point is far from the common center of primaries mass, however it is also periodic motion all time .
-1.0
0
5
10
15
20
t
Figure 10: Comparison among numerical, first and second approximated solutions at α0 = 0.6 (Color figure online).
More reading for the Figs. (5–10) the motion is periodic and the changes may be in the difference of its amplitude, that is considered the major difference between the obtained motion by the numerical solution and the other by the multiple scales solutions. It is also observed that when the infinitesimal body starts its motion from a point far from the common center of primaries mass, the obtained motion by the numerical may not be periodic, while a chaotic behaviour will appear in the obtained motion by the second approximated solution. The chaotic behaviour in motion is considered the major difference between the obtained motion by the first and second approximated solutions via multiple scales method. Regard to the multiple scale solutions, the motion is periodic all time, with regularity in its periodicity when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. While the motion may be chaotic when the infinitesimal body starts its motion from a point is far from the common center of primaries mass due to the second approximated solution, however it is periodic motion.
Now we summarize the status of motion regard to each solution in the following items: • The obtained motion by the first approximated solution is regular periodic motion all time whatever initial conditions. • The obtained motion by the numerical and the second approximated solutions is periodic when the infinitesimal body starts its motion from a nearer position to the common center of primaries mass. • When the infinitesimal body starts its motion from a point which is very far from the common center of primaries mass, the obtained motion by the numerical solution may not be periodic, while the obtained motion by the second approximated solution may be a chaotic, however it is also periodic all time. • The major difference between the obtained periodic motion by the numerical solution and the others by the approximated solutions is in the difference of its amplitude.
8. Conclusion In this paper two approximated analytical periodic solutions of Sitnikov problem have been developed. The multiple scales method is used to remove the secular terms and find periodic solutions in closed forms. At the same time of removing the secular terms the periodicity conditions for the approximated solutions are constructed.
• The major difference between the first and second approximated solutions of multiple scales method is a chaotic motion behaviour. Which will appear due to the provided solution by the second approximated solution when the infinitesimal body starts its motion from a point which is far from the common center of primaries mass.
Comparisons among numerical solutions (NS), the first approximated solutions (FA) and the second approximated solutions (SA) are investigated graphically in two main groups. In the first group a comparison among three issues for the same solution is developed by depending on a three different initial conditions. The different of these conditions means that the infinitesimal body starts its motion from three different positions from the common center of primaries mass. While in the second main group the
Finally we underline that the obtained solutions by multiple scales methods reflect the true motion of the Sitnikov restricted three–body problem, and the higher order approximation has more accuracy than the lower approximation. Moreover the solutions of multiple scales tech10
niques are more realistic than the numerical solution, because the numerical solution may not represent a periodic motion for a long time.
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