Periodic orbit in the frame work of restricted three bodies under the asteroids belt effect

Ahmed A. Abozaid; H. H. Selim; Kamel A. K. Gadallah; I. A. Hassan; Elbaz I. Abouelmagd

doi:10.2478/amns.2020.2.00022

Abstract

In this paper, we present a comprehensive analytical study on the perturbed restricted three bodies problem. We formulate the equations of motion of this problem, in the event of the asteroids belt perturbation. We find the locations of equilibrium points (collinear and triangular points) and analysis their linear stability. Furthermore the periodic orbits around both collinear and triangular points are found.

Keywords:: Restricted three bodies problem; Asteroids belt effect; Linear Stability; Periodic orbits; 37N05; 70F07; 70F15

1 Introduction

In general the model of three–body problem is related to the motion of three bodies, in space under mutual gravitational forces without restrictions or specified conditions. The importance of this model in celestial mechanics will rise when the three objects move in space under the effects of their mutual gravitational attractions. One of the most familiar emerged model from the general three–body problem is the restricted model. In this model, we impose that the third body, “infinitesimal body”, is very small than the other two bodices “primaries”, and it dose not affect their motion, the restricted model is called planer circular or elliptical restricted problem when the third body in moving in the plane of primaries motion [2,8,9,13,16,31], while is called spatial restricted three–body problem if the third body move in three dimensions [35].

In fact there are many issue of the “restricted three–body problem”, and that is regard to the existence of many disturbance forces. The studying of these issue enable us to get precise and accurate data about the dynamical features of the system. Which will have more significant particulary in space mission. The most important features of “restricted three–body problem” are the existence of libration points and their stability as well as the periodic motion around these points. There are many authors devoted their research to investigate the aforementioned properties within frame work of the “perturbed restricted three–body problem” [3, 5, 6, 10, 11, 15, 17, 33]. Furthermore, the analysis of lower or higher order of resonant periodic orbits with in frame of the photogravitational “restricted three–body problem” are studied by [28, 29].

In the frame work of studying the symmetric of periodic orbits, [27] analyzed the asymmetric solution in the restricted three–body problem. He investigated the symmetry of periodic orbits numerically. Moreover he use Levi–Civita transformation to regularize the equations of motion, in order to avoid the singularity between the third body and one of the primary bodies. [32] used theoretical and numerical approaches to investigate and study the symmetric relative periodic orbits within frame of the isosceles restricted problem three bodies. They also proved that the elastance of many families of symmetric relative periodic solution, which are emerged from heteroclinic connections between binary or triple collisions

[14] studied the real system of Saturn-Titan to explore the oblateness influence of Saturn planet on the periodic orbits and quasi-periodic motion regions around the primaries within frame restricted thee–body model. They analysed the positions, the quasi-periodic orbits and periodic size using the Poincaré surface of section technique. They proved that some quasi-periodic orbits change to periodic orbits corresponding the oblateness effect and vice-versa. [12] investigated also the periodic orbits around the libration pints, in the case of the bigger primary is radiating, while the smaller primary suffer from lack of sphericity, due to the effect of zonal harmonic coefficients, which are considered up to J₄. In addition [7] prove that the obtained first and second kind of periodic orbits of the unperturbed restricted 3–body problem can be extended to perturbed restricted 3–body problems, under the perturbed effect of the zonal harmonic coefficients and solar sail.

In the case of the primaries in the restricted model are enclitic by a ring-type belt of material particle points, the infinitesimal body motion is not valid, if we ignore the effect of this belt. Already in stellar systems there are rings of dust particles and asteroids belts around the planetary systems. Which are regarded as the young analogues of the Kuiper belt in our Solar System, see for more details [18]. Under the effect of asteroid belt, when the massive primaries are oblate and radiating, the locations of the equilibria points and the linear stability around these points are studied by [34]. They demonstrated that there are two new equilibrium points (L_n1 and L_n2) as well as the classical five points, which are found regard to the extra–gravitational asteroids belt effect.

The effect of the gravitational potential of the asteroids belt is not limited to the changes in the mathematical expressions, which represent the dynamical systems, but also its effect go to the dynamical properties of systems. This encouraged many researchers to study the dynamics of astronomical dynamical systems under the asteroids belt effect. For example, [20,21,22] investigated that the number and positions of equilibria, also showed that the solution curves topology will different, when the gravitational potential of asteroid belt is considered. They showed that the planetary system are affected by gravitational belt, where they proved that the probability to obtain equilibria points in the inner part of the belt is larger than to obtain near the outer part. The significant of their results is due to we can use it to investigate the observational configuration of Kuiper belt objects of the outer solar system.

[36] studied and analyzed a Chermnykh-like problem under the effect the gravitational potential of asteroid belt, and found a new equilibrium points for this problem. In addition the stability of equilibrium points when the smaller body is oblate spheroid and the bigger is a radiating body under the influence of the gravitational potential of asteroid belt, in the “restricted three–body problem” studied by [25]. The secular solution around the triangular equilibrium points when both massive bodies are oblate and radiating with the effect of asteroid belt are found and reduced to periodic one by [4] within frame restricted three–body problem.

In this paper we will study the perturbation of the gravitational potential of asteroid belt, which is constructed by [26] on the locations of the equilibrium and their stability as well as the periodic orbits around these points. This paper is organized as follow: An introduction, background on asteroids belt potential and a model descriptions are presented in Sections (1 – 3). While the locations of equilibrium points and there linear stability are studied in Sections (4 – 5). But the periodic orbits around these points are constructed in Section (6). Finally the conclusion is drawn in last Section.

2 Background on asteroids belt potential

In the solar system, the asteroid belt is similar to a ring-shaped. it can found between the Mars and Jupiter orbits. This region includes many objects (minor planets) with different sizes and shapes, which are irregular in most cases but very smaller than compered to the planets. In particularly, this belt is called the main asteroids belt, in order to characterize it from any other collection of asteroids in the solar system, such as trojan or near–earth asteroids, see Fig.1 (Source: https://en.wikipedia.org/wiki/Asteroidbelt). The asteroid belt region lies between the range of radial distances from 2.06 to 3.27 AU. It includes about 93.4% minor planets. These distances represent the inner and outer boundaries of the main belt region respectively [30]. The second law of motion and the universal gravitational law have been used as the most fundamental laws for the physical sciences, since their success in investigating the celestial bodies notion in the solar system. Thus the Newtonian Law was first proved in the astronomical context. It was then applied to other fields successfully. But the obtained results of this law lacks the accuracy in cases the of stellar or planetary systems have discs of dust or asteroids belt [19].

In the recent years, the researchers are studying the effect gravitational potential from a belt on the linear stability of libration points after was discovered dust ring around the star and discs around the planetary orbits [23,24]. There are perturbations in the solar system due to asteroid belt, where several of the largest asteroids are massive enough to significantly affect the orbits of other bodies for example affect the asteroids in the motion of Mars (Mars is very sensitive to perturbations from many minor planets), motion space probes affected by perturbation from asteroids and perturbations from asteroid on another asteroid when which close encounter.

In order to explore the orbital dynamics or the motion of the celestial dynamical systems, we have to build first suitable model that describing and realistically the structures and properties of the asteroid belt. One of the most important belt potential and used in the literatures introduced by Miyamoto-Nagai [26]. This model is called flattened potential and used in modelling disk galaxies. It can be controlled by

V_{b} (r, z) = \frac{M_{b}}{{(r^{2} + {(a + \sqrt{z^{2} + b^{2}})}^{2})}^{1 / 2}}

(1)

where

M_b is the total mass of the disc.
r is the radial distance of the infinitesimal body it is given by r² = x² + y².
The parameter a known as the flatness parameter determine the flatness of the profile.
The parameter b known as the core parameter determine the size of the core of density profile.

Hence the perturbed acceleration regard to the asteroid belt is a_b(r, z) = ∇V_b (r, z), thereby the acceleration can be written in the following form

a_{b} (r, z) = - \frac{M_{b}}{{(r^{2} + (a + \sqrt{z^{2} + b^{2}}))}^{3 / 2}} (x, y, z [1 + \frac{a}{\sqrt{z^{2} + b^{2}}}])

(2)

If a = b = 0 the potential reduces to the one by a point mass or spherical subject whose mass is M_b. Restricting ourselves to the XY–plane (z = 0), and define T ≡ a + b from Eq. (1) we have

V_{b} (r, 0) = \frac{M_{b}}{{(r^{2} + T^{2})}^{1 / 2}}

(3)

with a help of Eq. (3), we can get the acceleration in the XY–plane or substituting by z = 0 into Eq. (2), then one obtains

a_{b} (r, 0) = - \frac{M_{b} r}{{(r^{2} + T^{2})}^{3 / 2}} (x, y)

(4)

3 Model description

We assume that m₁ and m₂ denote the bigger and smaller primaries masses respectively, and m is the mass of the infinitesimal body. We consider both masses m₁ and m₂ move in circular orbits around their common center of mass. Furthermore the infinitesimal body m moves in the same plane of primaries motion under their mutual gravitational fields. We also assume that the coordinate system OXYZ rotates about OZ–axes by the angular velocity n in positive direction. OX–axis is taken the joining line between the primaries, OY – axis is perpendicular to OX–axis and OZ–axis is perpendicular to the orbital plane of the primaries. Let r₁ and r₂ be the distances between m and the primaries m₁ and m₂ respectively, while R the separation distance between m₁ and m₂. The coordinates of m₁, m₂ and m are (x₁,0,0), (x₂,0,0) and (x,y,0) respectively.

Now we normalize the units as the sum of two masses m₁ and m₂ is one and the distance between them also is taken as one. In addition the gravitational constant is one. We also assume that μ = m₂/(m₁ + m₂) be the mass parameter. Consequently m₂ = μ and m₁ = 1 − μ with m₁ > m₂ and 0 < μ ≤ 1/2. Then in the XY–plane, the coordinates of m₁,m₂ and m are (x₁,0,0) = (μ,0,0), (x₂,0,0) = (μ − 1,0,0) and (x,y,0) respectively. Consequently in the rotating coordinate dimensionless system, the motion equations of the infinitesimal body m under the gravitational potential of m₁ and m₂ are given by

\begin{matrix} \ddot{x} - 2 n \dot{y} = n^{2} x - \frac{(1 - μ)}{r_{1}^{3}} (x - μ) - \frac{μ}{r_{2}^{3}} (x - μ + 1) \\ \ddot{y} + 2 n \dot{x} = n^{2} y - \frac{(1 - μ) y}{r_{1}^{3}} - \frac{μ y}{r_{2}^{3}} \end{matrix}

(5)

In Eq. (5), the angular velocity n = 1, see for details [35], where

\begin{matrix} r_{1}^{2} = {(x - μ)}^{2} + y^{2} \\ r_{2}^{2} = {(x - μ + 1)}^{2} + y^{2} \end{matrix}

(6)

In the case of the gravitational potential of asteroids belt is considered, then with a help of Eq. (4), the perturbed dynamical system of the restricted three–body problem is controlled by

\begin{matrix} \ddot{x} - 2 n \dot{y} = Ω_{x} \\ \ddot{y} + 2 n \dot{x} = Ω_{y} \end{matrix}

(7)

where

Ω = \frac{n^{2} (x^{2} + y^{2})}{2} + \frac{1 - μ}{r_{1}} + \frac{μ}{r_{2}} + \frac{M_{b}}{{(r^{2} + T^{2})}^{1 / 2}}

(8)

and r₁ and r₂ are given by Eq. (6), while the perturbed mean motion n is

n^{2} = 1 + \frac{2 M_{b} r_{c}}{{(r_{c}^{2} + T^{2})}^{3 / 2}}

(9)

here in Eq. (9),

r_{c}^{2} = 1 - μ + μ^{2}

, see [34] for details.

Using Eqs. (7, 8) then the Jacobian integral is governed by

{\dot{x}}^{2} + {\dot{y}}^{2} - 2 Ω (x, y) + C = 0

In which C is the constant of integration.

4 Locations of equilibrium points

The equilibrium points are the locations of the infinitesimal body with zero velocity and zero acceleration, in the rotating reference frame. Then these locations can be found when

\ddot{x} = \ddot{y} = \dot{x} = \dot{y} = 0

, therefore Ω_x = Ω_y = 0, then we have to solve the following two equations

\begin{matrix} x [n^{2} - \frac{(1 - μ)}{r_{1}^{3}} - \frac{μ}{r_{2}^{3}} - \frac{M_{b}}{{(r^{2} + T^{2})}^{3 / 2}}] + μ (1 - μ) [\frac{1}{r_{1}^{3}} - \frac{1}{r_{2}^{3}}] = 0, \\ [n^{2} - \frac{(1 - μ)}{r_{1}^{3}} - \frac{μ}{r_{2}^{3}} - \frac{M_{b}}{{(r^{2} + T^{2})}^{3 / 2}}] y = 0 . \end{matrix}

(10)

Using the above two equations, we will investigate two cases for the locations of equilibrium points in the following subsection.

4.1 Location of collinear points

In the case of collinear equilibrium points (L₁, L₂ and L₃) y = 0, so that the equilibrium points lie on the line joining the primaries (X–axis), see Fig. 2, so we have

n^{2} x - \frac{(1 - μ) (x - μ)}{{| x - μ |}^{3}} - \frac{μ (x - μ + 1)}{{| x - μ + 1 |}^{3}} - \frac{M_{b} x}{{(r^{2} + T^{2})}^{3 / 2}} = 0

(11)

4.1.1 Location of L₁

The equilibrium point L₁ lies beyond masse m₁ as in Fig. 2. Then in this case, we have r₁ − r₂ = 1 therefor ∂r₁/∂x = ∂r₂/∂x = −1 and r₂ = μ − x − 1, r₁ = μ − x using Eq. (11), we get

n^{2} x + \frac{(1 - μ)}{{(x - μ)}^{2}} + \frac{μ}{{(x - μ + 1)}^{2}} - \frac{M_{b} x}{{(x^{2} + T^{2})}^{3 / 2}} = 0

(12)

Let r₂ = ξ₁, r₁ = 1 + ξ₁, x₁ = μ − 1 − ξ₁, and n² = s, then with a help of Eq. (12), we get

\begin{matrix} s (μ - 1 - ξ_{1}) + \frac{(1 - μ)}{{(1 + ξ_{1})}^{2}} + \frac{μ}{ξ_{1}^{2}} - \frac{M_{b}}{{(μ - 1 - ξ_{1})}^{2}} (1 - \frac{3}{2} \frac{T^{2}}{{(μ - 1 - ξ_{1})}^{2}}) = 0 \end{matrix}

(13)

after writing Eq. (13) in the series form, we get

\begin{matrix} \begin{matrix} μ + 6 μ ξ_{1} + ((1 - s - M_{b} + \frac{3 T^{2} M_{b}}{2}) + (10 + 5 s + 2 M_{b}) μ) ξ_{1}^{2} \\ + ((4 - 7 s - 4 M_{b} + 3 T^{2} M_{b}) + (4 + 30 s + 6 M_{b}) μ) ξ_{1}^{3} \\ + ((6 - 21 s - 6 M_{b} + \frac{3 T^{2} M_{b}}{2}) + (- 3 + 75 s + 6 M_{b}) μ) ξ_{1}^{4} \\ + ((4 - 35 s - 4 M_{b}) + (- 2 + 100 s + 2 M_{b}) μ) ξ_{1}^{5} \\ + ((1 - 35 s - M_{b}) + 75 s μ) ξ_{1}^{6} + (- 21 s + 30 s μ) ξ_{1}^{7} \\ + (- 7 s + 5 s μ) ξ_{1}^{8} - s ξ_{1}^{9} = 0 \end{matrix} \end{matrix}

Then the parameter μ as a function in the variation ξ₁ is given by

μ = a_{11} ξ_{1}^{2} + a_{12} ξ_{1}^{3} + a_{13} ξ_{1}^{4} + O {[ξ_{1}]}^{5}

where

\begin{matrix} a_{11} = - 1 + s + M_{b} - \frac{3 T^{2} M_{b}}{2} \\ a_{12} = 2 + s - 2 M_{b} + 6 T^{2} M_{b} \\ a_{13} = - 8 + 10 s - 5 s^{2} + 10 M_{b} - 7 s M_{b} - \frac{45 T^{2} M_{b}}{2} + \frac{15}{2} s T^{2} M_{b} \end{matrix}

(14)

Consequently, we will using the Lagrangian inversion method to inverting the above series, which represents μ, to express ξ₁ as functions of μ, then we get

ξ_{1} = c_{11} \sqrt{μ} + c_{12} μ + c_{13} μ^{3 / 2}

where

\begin{matrix} \begin{matrix} c_{11} = \sqrt{\frac{1}{- 1 + s + M_{b} - \frac{3 T^{2} M_{b}}{2}}} \\ c_{12} = \frac{(- 2 - s + 2 M_{b} - 6 T^{2} M_{b})}{(- 2 + 2 s + 2 M_{b} - 3 T^{2} M_{b}) (- 1 + s + M_{b} - \frac{3 T^{2} M_{b}}{2})} \\ c_{13} = (\frac{1}{2 {(- 2 + 2 s + 2 M_{b} - 3 T^{2} M_{b})}^{2}} {(\frac{1}{- 1 + s + M_{b} - \frac{3 T^{2} M_{b}}{2}})}^{3 / 2} \\ \times (\begin{matrix} - 12 + 92 s - 55 s^{2} + 20 s^{3} + 32 M_{b} - 128 s M_{b} \\ + 48 s^{2} M_{b} - 18 T^{2} M_{b} + 240 s T^{2} M_{b} - 60 s^{2} T^{2} M_{b} \end{matrix}) \end{matrix} \end{matrix}

(15)

hence

x_{1} = μ - 1 - c_{11} \sqrt{μ} - c_{12} μ - c_{13} μ^{3 / 2}

(16)

4.1.2 Location of L₂

Since the point L₂ lies between the two primaries, thereby r₁ + r₂ = 1, r₂ = x − μ + 1, r₁ = μ − x and ∂r₂/∂x = −∂r₁/∂x = 1, then by using Eq. (11), we get

n^{2} x + \frac{(1 - μ)}{{(x - μ)}^{2}} - \frac{μ}{{(x - μ + 1)}^{2}} - \frac{M_{b} x}{{(x^{2} + T^{2})}^{3 / 2}} = 0

(17)

If we take r₂ = ξ₂, r₁ = 1 − ξ₂ and x₂ = μ − 1 + ξ₂, then we can rewrite Eq. (17) in the following form

s (μ - 1 + ξ_{2}) - \frac{(1 - μ)}{{(1 - ξ_{2})}^{2}} - \frac{μ}{ξ_{2}^{2}} - \frac{M_{b}}{{(μ - 1 + ξ_{2})}^{2}} (1 - \frac{3}{2} \frac{T^{2}}{{(μ - 1 + ξ_{2})}^{2}}) = 0

(18)

again Eq. (18) in the series form is

\begin{matrix} \begin{matrix} - μ + 6 μ ξ_{2} + ((1 - M_{b} - s + 3 M_{b} T^{2} / 2) + (- 20 + 2 M_{b} + 5 s) μ) ξ_{2}^{2} \\ + ((- 4 + 4 M_{b} + 7 s - 3 M_{b} T^{2}) + (36 - 6 M_{b} - 30 s) μ) ξ_{2}^{3} \\ + ((6 - 6 M_{b} - 21 s + 3 M_{b} T^{2} / 2) + (- 33 + 6 M_{b} + 75 s) μ) ξ_{2}^{4} \\ + ((- 4 + 4 M_{b} + 35 s) + (14 - 2 M_{b} - 100 s) μ) ξ_{2}^{5} \\ + ((1 - M_{b} - 35 s) + (- 2 + 75 s) μ) ξ_{2}^{6} + (21 s - 30 s μ) ξ_{2}^{7} \\ + (- 7 s + 5 s μ) ξ_{2}^{8} + s ξ_{2}^{9} = 0 \end{matrix} \end{matrix}

Then the mass ratio μ as function in ξ₂ is given by

μ = - a_{11} ξ_{2}^{2} + a_{12} ξ_{2}^{3} + a_{23} ξ_{2}^{4} + O ξ_{2}^{5}

where a₁₁ and a₁₂ given by Eq. (14) while a₂₃ is given by

a_{23} = 2 + 2 s - s^{2} - 4 M_{b} + s M_{b} + \frac{39 T^{2} M_{b}}{2} - \frac{9}{2} s T^{2} M_{b}

again using the Lagrangian inversion method to inverting the above series, we get

ξ_{2} = - c_{11} \sqrt{μ} + c_{12} μ + c_{23} μ^{3 / 2} + O {[μ]}^{2}

where c₁₁ and c₁₂ given as Eq. (15) while c₂₃ is given by

\begin{matrix} c_{23} = (\frac{1}{2 {(- 2 + 2 s + 2 M_{b} - 3 T^{2} M_{b})}^{2}}) {(\frac{1}{1 - s - M_{b} + \frac{3 T^{2} M_{b}}{2}})}^{3 / 2} \\ \times (\begin{matrix} 12 + 20 s + 17 s^{2} - 4 s^{3} - 16 M_{b} - 32 s M_{b} \\ + 30 T^{2} M_{b} + 144 s T^{2} M_{b} - 12 s^{2} T^{2} M_{b} \end{matrix}) μ^{3 / 2}) \end{matrix}

hence

x_{2} = μ - 1 + c_{21} \sqrt{μ} + c_{22} μ + c_{23} μ^{3 / 2} + O {[μ]}^{2}

(19)

4.1.3 Location of L₃

The point L₃ lies beyond the large mass according to Fig. 2, r₂ − r₁ = 1, r₁ = x − μr₂ = x − μ + 1 and ∂r₁/∂x = ∂r₂/∂x = 1, then Eq. (11) can be rewritten in the form

n^{2} x - \frac{(1 - μ)}{{(x - μ)}^{2}} - \frac{μ}{{(x - μ + 1)}^{2}} - \frac{M_{b} x}{{(x^{2} + T^{2})}^{3 / 2}} = 0

(20)

after substituting r₂ = ξ₃ + 2 and r₁ = ξ₃ + 1, into Eq. (20), we get

s (μ + ξ_{3} + 1) - \frac{(1 - μ)}{{(1 + ξ_{3})}^{2}} - \frac{μ}{{(2 + ξ_{3})}^{2}} - \frac{M_{b}}{{(μ + ξ_{3} + 1)}^{2}} (1 - \frac{3}{2} \frac{T^{2}}{{(μ + ξ_{3} + 1)}^{2}}) = 0

(21)

Again the form series of Eq. (21) is given by

\begin{matrix} a_{30} + d_{30} μ + (a_{31} + d_{31} μ) ξ + (a_{32} + d_{32} μ) ξ_{3}^{2} + (a_{33} + d_{33} μ) ξ_{3}^{3} \\ + (a_{34} + d_{34} μ) ξ_{3}^{4} + (a_{35} + d_{35} μ) ξ_{3}^{5} + (a_{36} + d_{36} μ) ξ_{3}^{6} \\ + (a_{37} + d_{37} μ) ξ_{3}^{7} + (a_{38} + d_{38} μ) ξ_{3}^{8} + (a_{39} + d_{39} μ) ξ_{3}^{9} = 0 \end{matrix}

where

\begin{matrix} a_{30} = - 1 - M_{b} + s + \frac{3 M_{b} T^{2}}{2}, & d_{30} = \frac{3}{4} + 2 M_{b} + s - 6 M_{b} T^{2} \\ a_{31} = 2 + 2 M_{b} + s - 6 M_{b} T^{2}, & d_{31} = - \frac{7}{4} - 6 M_{b} + 30 M_{b} T^{2} \\ a_{32} = - 3 - 3 M_{b} + 15 M_{b} T^{2}, & d_{32} = \frac{45}{16} + 12 M_{b} - 90 M_{b} T^{2} \\ a_{33} = (4 + 4 M_{b} - 30 M_{b} T^{2}, & d_{33} = - \frac{31}{8} - 20 M_{b} + 210 M_{b} T^{2} \\ a_{34} = - 5 - 5 M_{b} + \frac{105 M_{b} T^{2}}{2}, & d_{34} = \frac{315}{64} + 30 M_{b} - 420 M_{b} T^{2} \\ a_{35} = 6 + 6 M_{b} - 84 M_{b} T^{2}, & d_{35} = - \frac{381}{64} - 42 M_{b} + 756 M_{b} T^{2} \\ a_{36} = - 7 - 7 M_{b} + 126 M_{b} T^{2}, & d_{36} = \frac{1785}{256} + 56 M_{b} - 1260 M T^{2} \\ a_{37} = 8 + 8 M_{b} - 180 M_{b} T^{2}, & d_{37} = - \frac{511}{64} - 72 M_{b} + 1980 M_{b} T^{2} \\ a_{38} = - 9 - 9 M_{b} + \frac{495 M_{b} T^{2}}{2}, & d_{38} = \frac{9207}{1024} + 90 M_{b} - 2970 M_{b} T^{2} \\ a_{39} = 10 + 10 M - 330 M_{b} T^{2}, & d_{39} = - \frac{10235}{1024} - 110 M_{b} + 4290 M_{b} T^{2} \end{matrix}

Then μ as a function series in ξ₃ is given by

μ = b_{30} + b_{31} ξ_{3} + b_{32} ξ_{3}^{2} + O [ξ_{3}^{3}]

where

\begin{matrix} b_{30} = \frac{32 (- 2 - 2 M_{b} + 2 s + 3 M T^{2})}{- 48 - 128 M_{b} - 64 s + 384 M_{b} T^{2}} \\ b_{31} = \frac{2 (2 + 18 M_{b} + 16 M_{b}^{2} - 36 s - 80 M_{b} s - 8 s^{2} - 129 M_{b} T^{2} + 336 M_{b} s T^{2})}{{(- 3 - 8 M_{b} - 4 s + 24 M_{b} T^{2})}^{2}} \\ b_{32} = \frac{1}{2 {(- 3 - 8 M_{b} - 4 s + 24 M_{b} T^{2})}^{3}} (\begin{matrix} - 2 + 78 M_{b} + 522 s + 3608 M_{b} s - 520 s^{2} - 1152 M_{b} s^{2} \\ - 2097 M_{b} T^{2} - 20460 M_{b} s T^{2} + 9600 M_{b} s^{2} T^{2} \end{matrix}) \end{matrix}

Now using the Lagrangian inversion method to inverting the above series, we get

ξ_{3} = c_{31} μ + c_{32}

where

\begin{matrix} c_{31} = \frac{{(- 3 - 8 M_{b} - 4 s + 24 M_{b} T^{2})}^{2}}{2 (2 + 18 M_{b} - 36 s - 80 M_{b} s - 8 s^{2} - 129 M_{b} T^{2})} \\ c_{32} = \frac{{(- 3 - 8 M_{b} - 4 s + 24 M_{b} T^{2})}^{2} (- \frac{32 (- 2 - 2 M_{b} + 2 s + 3 M_{b} T^{2})}{- 48 - 128 M_{b} - 64 s + 384 M_{b} T^{2}})}{2 (2 + 18 M_{b} - 36 s - 80 M_{b} s - 8 s^{2} - 129 M_{b} T^{2} + 336 M_{b} s T^{2})} \end{matrix}

hence

x_{3} = 1 + μ + c_{31} μ + c_{32}

(22)

Finally we can use Eqs. (16, 19, 22) to find the locations of collinear points.

4.2 Location of triangular points

In the case of the triangular equilibrium points (L₄ and L₅) y ≠ 0 and Ω_x = 0 = Ω_y. Using Eqs. (10), we get

\begin{matrix} n^{2} - \frac{(1 - μ)}{r_{1}^{3}} - \frac{μ}{r_{2}^{3}} - \frac{M_{b}}{{(r^{2} + T^{2})}^{3 / 2}} = 0 \\ \frac{μ (1 - μ)}{r_{1}^{3}} - \frac{μ (1 - μ)}{r_{2}^{3}} = 0 \end{matrix}

(23)

From the second equation in Eq. (23), we get

r_{1} = r_{2} .

If we assume that the effect of the potential from the belt is neglected, i.e., M_b = 0, then Eqs. (23) are reduced to the classical case of Szebehely solutions r₁ = r₂ = 1 [35]. But, due to the perturbations from the belt, we assume that the solutions may change slightly by ɛ, then we can write

r_{1} = r_{2} = 1 + ε

(24)

From Eqs. (9, 23, 24), we get

1 - {(1 + ε)}^{- 3} + \frac{2 M_{b} r_{c}}{{(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{M_{b}}{{(r^{2} + T^{2})}^{3 / 2}} = 0

(25)

where ɛ is very small quantities represents perturbation effect of asteroids belt.

Now we will keep the linear terms ɛ and neglecting the higher orders, then with a help of Eq. (25), we have

ε = - \frac{M_{b} (2 r_{c} - 1)}{3 {(r_{c}^{2} + T^{2})}^{3 / 2}}

(26)

From Eqs. (6, 24, 26), we get the locations of the triangular points in the following form

x = μ - \frac{1}{2}, y = \pm \frac{\sqrt{3}}{2} (1 - \frac{4 M_{b} (2 r_{c} - 1)}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}})

(27)

5 Stability of motion around the libration point

After determining the locations of libration points, we will move to understand the stability motion properties around these points. In order to study the motion of the infinitesimal body in the neighborhood of an equilibrium points (x₀, y₀), we employ small displacement (ξ, η) to the coordinate (x₀, y₀) where (x₀, y₀) represents the coordinates of one of five equilibria points. So that the vector of variation is related to the initial stat vector by r = r₀ + Δr where r₀ ≡ (x₀, y₀), r ≡ (x, y) and Δr ≡ (ξ, η), then we can write

\begin{matrix} x = x_{0} + ξ \\ y = y_{0} + η \end{matrix}

(28)

We linearize the equations of motion by using Taylor series around equilibrium. Then using Eqs. (7) and Eqs. (28), we obtain

\begin{matrix} \ddot{ξ} - 2 n \dot{η} = Ω_{x}^{0} + \frac{1}{1!} (ξ \frac{\partial}{\partial x} + η \frac{\partial}{\partial y}) Ω_{x}^{0} + \frac{1}{2!} {(ξ \frac{\partial}{\partial x} + η \frac{\partial}{\partial y})}^{2} Ω_{x}^{0} + O (3) \\ \ddot{η} - 2 n \dot{ξ} = Ω_{y}^{0} + \frac{1}{1!} (ξ \frac{\partial}{\partial x} + η \frac{\partial}{\partial y}) Ω_{y}^{0} + \frac{1}{2!} {(ξ \frac{\partial}{\partial x} + η \frac{\partial}{\partial y})}^{2} Ω_{y}^{0} + O (3) \end{matrix}

From the above equations the linear variational equations are

\begin{matrix} \ddot{ξ} - 2 n \dot{η} = ξ Ω_{xx}^{0} + Ω_{xy}^{0} η \\ \ddot{η} + 2 n \dot{ξ} = ξ Ω_{xy}^{0} + Ω_{yy}^{0} η \end{matrix}

(29)

where subscripts x and y denoted to the second partial derivatives of Ω. While the superscript 0 indicates that the partial derivatives have been evaluated at one of the equilibrium points (x₀, y₀), where these derivatives are given by

\begin{matrix} Ω_{xx} = n^{2} + \frac{3 (1 - μ) {(x - μ)}^{2}}{r_{1}^{5}} - \frac{1 - μ}{r_{1}^{3}} + \frac{3 μ {(x + 1 - μ)}^{2}}{r_{2}^{5}} - \frac{μ}{r_{2}^{3}} - \frac{M_{b}}{{(r^{2} + T^{2})}^{3 / 2}} + \frac{3 M_{b} x^{2}}{{(r^{2} + T^{2})}^{5 / 2}} \\ Ω_{yy} = n^{2} + \frac{3 (1 - μ) y^{2}}{r_{1}^{5}} - \frac{1 - μ}{r_{1}^{3}} + \frac{3 μ y^{2}}{r_{2}^{5}} - \frac{μ}{r_{2}^{3}} - \frac{M_{b}}{{(r^{2} + T^{2})}^{3 / 2}} + \frac{3 M_{b} y^{2}}{{(r^{2} + T^{2})}^{5 / 2}} \\ Ω_{xy} = \frac{3 (1 - μ) (x - μ) y}{r_{1}^{5}} + \frac{3 μ (x + 1 - μ) y}{r_{2}^{5}} + \frac{3 M_{b} x y}{{(r^{2} + T^{2})}^{5 / 2}} \end{matrix}

(30)

Now we suppose that the solutions of the Eqs. (29) are

ξ = {Ke}^{λ t}, η = {Me}^{λ t}

(31)

where K, M and λ constant, thereby

\dot{ξ} = k λ e^{λ t}, \ddot{ξ} = k λ^{2} e^{λ t}, \dot{η} = M λ e^{λ t}, \ddot{η} = M λ^{2} e^{λ t} .

(32)

substituting Eqs. (31, 32) into Eqs. (29), then the characteristic equation is determined by

| \begin{matrix} λ^{2} - Ω_{xx} & - 2 n λ - Ω_{xy} \\ 2 n λ - Ω_{xy} & λ^{2} - Ω_{yy} \end{matrix} | = 0

hence, we get

λ^{4} + (4 n^{2} - Ω_{xx}^{0} - Ω_{yy}^{0}) λ^{2} + Ω_{xx}^{0} Ω_{yy}^{0} - {(Ω_{xy}^{0})}^{2} = 0

(33)

The equilibrium point (x₀, y₀) is stable, when all the roots of the characteristic Eq. (33) are distinct pure imaginary numbers, this will be search in the next section.

5.1 Stability of collinear points

At the collinear libration points y = 0, and r₁ = |x − μ| and r₂ = |x − μ + 1|. From Eqs. (30, 33), the characteristic equation in case of collinear libration points can be reduced to

λ^{4} + (4 n^{2} - Ω_{xx}^{0} - Ω_{yy}^{0}) λ^{2} + Ω_{xx}^{0} Ω_{yy}^{0} = 0

(34)

Let Λ = λ², Eq. (34) can be rewritten in the form

Λ^{2} + b_{c} Λ + c_{c} = 0

(35)

then Eq. (35) represents a quadratic equation, which its solution can be written in the form

Λ_{1,2} = - \frac{1}{2} [b_{c} \pm \sqrt{b_{c}^{2} - 4 c_{c}}]

(36)

where

\begin{matrix} b_{c} = 4 n^{2} - Ω_{xx}^{0} - Ω_{yy}^{0} \\ c_{c} = Ω_{xx}^{0} Ω_{yy}^{0} \end{matrix}

(37)

Then with a help of Eqs. (36, 37), the roots of characteristic Eq. (34) can be write as λ_1,2 = ±σ, λ_3,4 = ±iτ where σ and τ are real number, which can be calculated by

\begin{matrix} σ^{2} = \frac{1}{2} [\sqrt{b_{c}^{2} - 4 c_{c}} - b_{c}] \\ τ^{2} = \frac{1}{2} [\sqrt{b_{c}^{2} - 4 c_{c}} + b_{c}] \end{matrix}

From the above two equations, we can determine the eigenvalues and the properties of the equilibrium point.

5.2 Stability of triangular points

At the triangular points, we have

\begin{matrix} Ω_{xx}^{0} = \frac{3}{4} (1 + \frac{5 M_{b} (2 r_{c} - 1)}{3 {(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{M_{b} {(μ - 1 / 2)}^{2}}{3 {(r_{c}^{2} + T^{2})}^{5 / 2}}) \\ Ω_{yy}^{0} = \frac{9}{4} (1 + \frac{7 M_{b} (2 r_{c} - 1)}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) \\ Ω_{xy}^{0} = \mp \frac{3 \sqrt{3}}{4} (1 - 2 μ) (1 + \frac{M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}} - \frac{11 M_{b} (2 r_{c} - 1)}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}}) \end{matrix}

(38)

and substituting λ² = ω into the characteristic Eq. (33), then we get

ω^{2} + b ω + c = 0

(39)

where a and c can be evaluated with a help of Eqs. (38) and the following two relations

\begin{matrix} b = 4 n^{2} - Ω_{xx}^{0} - Ω_{yy}^{0} \\ c = Ω_{xx}^{0} Ω_{yy}^{0} - {(Ω_{xy}^{0})}^{2} \end{matrix}

hence we have

\begin{matrix} b = 1 + \frac{M_{b} (2 r_{c} + 3)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{3 M_{b} r_{c}^{2}}{{(r_{c}^{2} + T^{2})}^{5 / 2}} \\ c = (- \frac{27}{4} - \frac{33 M_{b} (2 r_{c} - 1)}{2 {(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{27 M_{b}}{4 {(r_{c}^{2} + T^{2})}^{5 / 2}}) μ^{2} \\ + (\frac{27}{4} + \frac{33 M_{b} (2 r_{c} - 1)}{2 {(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{27 M_{b}}{4 {(r_{c}^{2} + T^{2})}^{5 / 2}}) μ \end{matrix}

(40)

Thereby the roots of Eq. (39) are given by

ω_{1,2} = - \frac{1}{2} [b \mp \sqrt{D}]

(41)

where D = b² − 4c is discriminant

\begin{matrix} D = (27 + \frac{66 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{27 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) μ^{2} \\ - (27 + \frac{66 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{27 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) μ \\ + 1 + \frac{2 M_{b} (2 r_{c} + 3)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{6 M_{b} r_{c}^{2}}{{(r_{c}^{2} + T^{2})}^{5 / 2}} \end{matrix}

(42)

which can be written as of a function of the mass parameter μ in the form

D = α μ^{2} - β μ + γ

(43)

here α and β are the coefficients μ² and μ respectively. Using Eq. (41) the roots are given by

λ_{1,2} = \pm \sqrt{ω_{1}}, λ_{3,4} = \pm \sqrt{ω_{2}}

Since 0 < μ ≤ 1/2, then with using Eq. (43), we can study the behavior of D in the interval (0, 1/2)

D = {\begin{matrix} 1 + (\frac{2 M_{b} (2 r_{c} + 3)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{6 M_{b} r_{c}^{2}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) > 0, when μ = 0 \\ - (\frac{23}{4} + \frac{M_{b} (58 r_{c} - 45)}{2 {(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{3 M_{b} (8 r_{c}^{2} + 9)}{4 {(r_{c}^{2} + T^{2})}^{5 / 2}}) < 0, when μ = 1 / 2 \end{matrix}

(44)

Also the derivative of D with respect to the parameter μ is given by

\begin{matrix} \frac{dD}{d μ} = 2 (27 + \frac{66 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{27 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) μ \\ - (27 + \frac{66 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{27 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) \end{matrix}

thereby

\frac{dD}{d μ} = {\begin{matrix} - 27 - (\frac{66 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{27 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) < 0, & when μ = 0 \\ 0, & when μ = 1 / 2 \end{matrix}

(45)

because M_b ≪ 1, and using Eqs. (44, 45), we obtain

\frac{dD}{d μ} \leq 0 \forall μ \in (0,1 / 2)

Eqs. (44, 45) show that the discriminant D has two different signs at the end of interval (0, 1/2), further dD/dμ < 0 in the interval (0,−β / 2α). Then D is strictly decreasing function in this interval, and there is only one value for μ in (0, 1/2), where D vanish, which is called the critical mass parameter (μ_c). Consequently we will examine three possible cases for the value of μ.

If 0 < μ < μ_c implies D = b² −4c > 0, and D decreasing in the interval (0, 1/2). Since b > 0, $b^{2} - 4 c < b^{2} (\sqrt{b^{2} - 4 c} < b)$ then ω < 0, thereby the four roots of λ are distinct pure imaginary numbers. Hence the triangular points are stable in this interval.
If μ = μ_c (D = 0), then we have double equal roots of λ which lead to secular terms, thereby the triangular points are unstable.
When μ_c < μ < 1/2, then D < 0 and we obtain four complex roots, with two of them whose the same real part and positive. Therefore the triangular points are also unstable.

5.3 Critical mass

Under the previous discussion, when Eq. (43) is equal zero, then one can obtain the value of critical mass (μ_c), which is governed by

μ_{c} = - \frac{1}{2 α} [α + \sqrt{α^{2} - 4 α γ}]

(46)

where

\begin{matrix} α^{2} = 729 + \frac{3564 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{1458 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}} \\ 4 α γ = 108 + \frac{216 M_{b} (2 r_{c} + 3)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{648 M_{b} r_{c}^{2}}{{(r_{c}^{2} + T^{2})}^{5 / 2}} \\ + \frac{264 M_{b} (2 r_{c} - 1)}{{(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{108 M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}} \end{matrix}

(47)

utilizing Eqs. (46, 47) we get

μ_{c} = μ_{0} + μ_{p}

(48)

Eq. (48) represents the value of the critical mass with two parts. The first μ₀) is related to the classical restricted problem without perturbation. While the second (μ_p) is related to the effect of asteroids belt. Of course in the absence of the asteroids belt effect, the value of critical mass is given by μ_c = μ₀, where the values of μ₀ and μ_p are given by the following relations

μ_{0} = \frac{1}{2} (1 - \frac{\sqrt{69}}{9}), μ_{p} = \frac{(76 - 8 r_{c}) M_{b}}{27 \sqrt{69} {(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{(1 + 6 r_{c}^{2}) M_{b}}{3 \sqrt{69} {(r_{c}^{2} + T^{2})}^{5 / 2}}

6 Periodic orbits

6.1 Periodic orbits around collinear points

Now it is easy to obtain the periodic orbits around the collinear points. Although these points are unstable i.e. if a body in any of these points is disturbed, a body will move a way. After substituting Eq. (31) into Eq. (29) with some simple computations, we will get a relation between the coefficients K_j and M_j, it is governed by

M_{j} = α_{j} K_{j}

(49)

where

α_{j} = \frac{(λ_{j}^{2} - Ω_{xx}^{0})}{2 n λ_{j}} = \frac{2 n λ_{j}}{Ω_{yy}^{0} - λ_{j}^{2}}

This relation means that the coefficients K_j and M_j (j = 1, 2, 3, 4) are dependent. Therefore the four initial conditions ξ₀, η₀,

{\dot{ξ}}_{0}

and

{\dot{η}}_{0}

associated with Eqs. (29) will determine the two sets of coefficients and will completely determine the eight coefficients (K_j, M_j). Where the subscript 0 indicates to these quantities are evaluated at the initial time (t = t₀). Substituting Eq. (49) into Eq. (31), we get

\begin{matrix} ξ_{0} = \sum_{j = 1}^{4} K_{j} e^{λ_{j} t} & {\dot{ξ}}_{0} = \sum_{j = 1}^{4} K_{j} λ_{j} e^{λ_{j} t} \\ η_{0} = \sum_{j = 1}^{4} K_{j} α_{j} e^{λ_{j} t} & {\dot{η}}_{0} = \sum_{j = 1}^{4} K_{j} λ_{j} α_{j} e^{λ_{j} t} \end{matrix}

(50)

The coefficient can be expressed as function of the initial conditions, because the determinant (Δ) of System of Eqs. (50) is not zero.

Δ = - \sqrt{\frac{Ω_{xx}^{0}}{Ω_{yy}^{0}}} [{(Ω_{xx}^{0} + Ω_{yy}^{0} - 4 n)}^{2} - \frac{1}{2} Ω_{xx}^{0} Ω_{yy}^{0}] \neq 0

The motion is bounded and consists of two harmonic motion when all roots of characteristic equation are purely imaginary [1], where the solution depends on the eigenfrequencies σ, τ, can be written in the from

\begin{matrix} ξ = K_{1} cos σ (t - t_{0}) + K_{2} sin σ (t - t_{0}) + K_{3} cos τ (t - t_{0}) + K_{4} sin τ (t - t_{0}), \\ η = M_{1} cos σ (t - t_{0}) + M_{2} sin σ (t - t_{0}) + M_{3} cos τ (t - t_{0}) + M_{4} sin τ (t - t_{0}) . \end{matrix}

(51)

Now We can take K₁ = K₂ = 0, therefor the solution in Eqs. (51) can rewritten in the following form

\begin{matrix} ξ = ξ_{0} cos τ (t - t_{0}) + \frac{η_{0}}{β_{3}} sin τ (t - t_{0}) \\ η = η_{0} cos τ (t - t_{0}) - ξ_{0} β_{3} sin τ (t - t_{0}) \end{matrix}

(52)

where the real quantities τ and β₃ are defined by

\begin{matrix} τ = {\frac{1}{2} [\sqrt{{(4 n^{2} - Ω_{xx}^{0} - Ω_{yy}^{0})}^{2} - 4 Ω_{xx}^{0} Ω_{yy}^{0}} + 4 n^{2} - Ω_{xx}^{0} - Ω_{yy}^{0}]}^{1 / 2} \\ β_{3} = \frac{τ^{2} + Ω_{xx}^{0}}{2 n τ} = \frac{2 n τ}{τ^{2} + Ω_{yy}^{0}} \end{matrix}

in which λ₃ = iτ and α₃ = iβ₃

From Eqs. (52) we get the velocity variation in the form

\begin{matrix} \dot{ξ} = - ξ_{0} τ sin τ (t - t_{0}) + \frac{η_{0}}{β_{3}} τ cos τ (t - t_{0}) \\ \dot{η} = - η_{0} τ sin τ (t - t_{0}) - ξ_{0} β_{3} τ cos τ (t - t_{0}) \end{matrix}

(53)

Using Eqs. (53), when t = t₀, one obtains

{\dot{ξ}}_{0} = η_{0} τ / β_{3}, {\dot{η}}_{0} = - ξ_{0} β_{3} τ

(54)

With a help of Eq. (54) we can eliminate the time from Eq. (52), and the equation of periodic orbits reduced to

ξ^{2} β_{3}^{2} + η^{2} = η_{0}^{2} + ξ_{0}^{2} β_{3}^{2}

which can be rewritten in the following standard form

\frac{ξ^{2}}{(η_{0}^{2} + ξ_{0}^{2} β_{3}^{2}) / β_{3}^{2}} + \frac{η^{2}}{(η_{0}^{2} + ξ_{0}^{2} β_{3}^{2})} = 1

(55)

Eq. (64) shows that the trajectory of the body around the collinear points is an ellipses whose center at these points. The parameters of the ellipse, the semi–major axes (a_c), the semi-minor axes (b_c) and the eccentricity (e_c) are given by

a_{c}^{2} = (η_{0}^{2} + ξ_{0}^{2} β_{3}^{2}), a_{c}^{2} = (η_{0}^{2} + ξ_{0}^{2} β_{3}^{2}), e_{c}^{2} = (1 - \frac{1}{β_{3}^{2}})

where the semi–major axes parallel to the η–axes while the semi-minor axes parallel to the ξ–axes. Also the periodic time (T_c) can be calculated by T_c = 2π/τ. Since

{\dot{η}}_{0} = - ξ_{0} β_{3} τ

,

{\dot{ξ}}_{0} = 0

at ξ₀≠ 0 and η₀ = 0, the motion along the orbits is retrograde.

6.2 Periodic orbits around triangular points

The triangular points are linearly stable in the range 0 < μ < μ_c. And the characteristic equation has four purely imaginary roots in neighborhood of the triangular points. So we have bounded motion around the triangular points. Which composed of two harmonic motions governed by the variation ξ and η by the following relations

\begin{matrix} ξ = C_{1} cos s_{1} t + D_{1} sin s_{1} t + C_{2} cos s_{2} t + D_{2} sin s_{2} t, \\ η = {\bar{C}}_{1} cos s_{1} t + {\bar{D}}_{1} sin s_{1} t + {\bar{C}}_{2} cos s_{2} t + {\bar{D}}_{2} sin s_{2} t . \end{matrix}

(56)

Therefor the terms with coefficients C₁, D₁,

{\bar{C}}_{1}

,

{\bar{D}}_{1}

associated with angular frequencies s₁ (mean motion) refer to the long periodic orbits and terms with coefficients C₂, D₂,

{\bar{C}}_{2}

,

{\bar{D}}_{2}

associated with angular frequencies s₂, which refer to the short periodic orbits. In addition

s_{1,2}^{2} = - ω

, thereby we get

s_{1,2}^{2} = \frac{1}{2} [b \mp \sqrt{D}]

(57)

Substituting Eqs. (42, 40) into Eq. (57), with some simple calculations, the angular frequencies of motion s₁ and s₂ are given by

\begin{matrix} s_{1} = \frac{3 \sqrt{3}}{2} \sqrt{μ (1 - μ)} (1 + \frac{11 M_{b} (2 r_{c} - 1)}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{M_{b}}{2 {(r_{c}^{2} + T^{2})}^{5 / 2}}) \\ s_{2} = 1 - \frac{27}{8} μ (1 - μ) (1 + \frac{22 M_{b} (2 r_{c} - 1)}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}} + \frac{M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}}) \\ + \frac{M_{b} (2 r_{c} + 3)}{2 {(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{3 M_{b} r_{c}^{2}}{2 {(r_{c}^{2} + T^{2})}^{5 / 2}} \end{matrix}

(58)

We can determine the relation between the coefficients of short and long period terms, when we substituting Eq. (56) into Eq. (29) and equating the coefficients of sine and cosine terms, we get

\begin{matrix} {\bar{C}}_{i} = Γ_{i} (2 n D_{i} s_{i} - Ω_{xy}^{0} C_{i}) \\ {\bar{D}}_{i} = - Γ_{i} (2 n C_{i} s_{i} + Ω_{xy}^{0} D_{i}) \end{matrix}

where

Γ_{i} = \frac{1}{s_{i}^{2} + Ω_{yy}^{0}} = \frac{s_{i}^{2} + Ω_{xx}^{0}}{4 n^{2} s_{i}^{2} + {(Ω_{xy}^{0})}^{2}} > 0

here (i = 1, 2) and

Ω_{xx}^{0}

,

Ω_{xy}^{0}

and

Ω_{yy}^{0}

are given by Eqs. (38).

Either the long or short period terms can be eliminated from the solving by properly selected initial conditions. The four initial conditions at t = 0 (ξ₀, η₀,

{\dot{ξ}}_{0}

and

{\dot{η}}_{0}

) are linearly related to the four coefficient. Therefore we are not found difficulty in establishing the desired initial condition. Hence if we suppose that the short periodic terms are vanished, i.e.

C_{2} = D_{2} = {\bar{C}}_{2} = {\bar{D}}_{2} = 0

, then the relation between the initial conditions and the coefficients in Eqs. (56) are

ξ_{0} = C_{1}, η_{0} = {\bar{C}}_{1}, {\dot{ξ}}_{0} = D_{1} s_{1}, {\dot{η}}_{0} = {\bar{D}}_{1} s_{1}

where

D_{1} = \frac{ξ_{0} Ω_{xy}^{0} + η_{0} (Ω_{yy}^{0} + s_{1}^{2})}{2 n s_{1}}, {\bar{D}}_{1} = \frac{ξ_{0} (Ω_{xx}^{0} + s_{1}^{2}) + η_{0} Ω_{xy}^{0}}{2 n s_{1}}

6.2.1 Elliptic orbits

Now we assume that a triangular point represents the origin of the coordinates system, where the third body starts its motion at the origin of the coordinate system. So we can get the initial conditions from Eq. (27) by (ξ₀, η₀) = (−x₀, −y₀) where

ξ_{0} = \frac{1}{2} - μ, η_{0} = \mp \frac{\sqrt{3}}{2} (1 - \frac{4 M_{b} (2 r_{c} - 1)}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}})

In which the negative sign (plus) means that the infinitesimal body starts its motion from L₄ (L₅). The trajectory of infinitesimal body after the elimination the short or long periodic terms becomes an ellipse. Which can be seen when we rewritten Eq. (56) for the long periodic solution in the form

\begin{matrix} ξ = C_{1} cos s_{1} t + D_{1} sin s_{1} t, \\ η = {\bar{C}}_{1} cos s_{1} t + {\bar{D}}_{1} sin s_{1} t . \end{matrix}

(59)

After eliminating the time from Eqs. (59), we get

α_{1} ξ^{2} + η^{2} + 2 β_{1} ξ η = γ_{1}

(60)

Since

| \begin{matrix} Γ_{1}^{2} (4 n^{2} s_{1}^{2} + {(Ω_{xy}^{0})}^{2}) & Γ_{1} Ω_{xy}^{0} \\ Γ_{1} Ω_{xy}^{0} & 1 \end{matrix} | = {(2 n s_{1} Γ_{1})}^{2} > 0

Eq. (60) represents of an ellipse with center at the origin coordinate system (ξ, η), which is coinciding with L₄ or L₄, where

α_{1} = \frac{4 n^{2} s_{1}^{2} + {(Ω_{xy}^{0})}^{2}}{{(s_{1}^{2} + Ω_{yy}^{0})}^{2}}, β_{1} = \frac{Ω_{xy}^{0}}{s_{1}^{2} + Ω_{yy}^{0}}, γ_{1} = α_{1} ξ_{0}^{2} + 2 β_{1} ξ_{0} η_{0} + η_{0}^{2}

6.2.2 The orientation of principal axes of the ellipse

Since Eq. (60) includes bilinear term ξ η that appears as a result of the rotation of the principal axes of ellipses through an angle θ with respect to the coordinate system (ξ, η). So we introduce a new coordinate reference frame

(\bar{ξ}, \bar{η})

called normal coordinates such that the bilinear term dose not appear. Hence the old and new coordinates system are related by the following equation

\begin{matrix} ξ = \bar{ξ} cos θ - \bar{η} sin θ \\ η = \bar{ξ} sin θ + \bar{η} cos θ \end{matrix}

(61)

Substituting Eqs. (61) into Eq. (60) and equate the coefficient of

\bar{ξ} \bar{η}

by zero, therefor after simplify the equation, we get

\frac{{\bar{ξ}}^{2}}{{\bar{a}}^{2}} + \frac{{\bar{η}}^{2}}{{\bar{b}}^{2}} = 1

(62)

where the orientation of the principal axes is governed by

tan 2 θ = \frac{2 Ω_{xy}^{0}}{Ω_{xx}^{0} - Ω_{yy}^{0}}

(63)

From Eqs. (38, 63), we obtain

tan 2 θ = \pm \sqrt{3} (1 - 2 μ + \frac{8 (1 - 2 μ) (2 r_{c} - 1) M_{b}}{9 {(r_{c}^{2} + T^{2})}^{3 / 2}} - \frac{2 μ (1 - 3 μ) M_{b}}{{(r_{c}^{2} + T^{2})}^{5 / 2}})

where plus sign (minus sign) refers to the center of ellipse at L₄ (L₅).

Furthermore the lengths of semi–major (a), and semi–minor (b) axes are controlled by

\begin{matrix} {\bar{a}}^{2} = \frac{2 γ_{1}}{((1 + α_{1}) - (1 - α_{1}) cos 2 θ + 2 β_{1} sin 2 θ)} \\ {\bar{b}}^{2} = \frac{2 γ_{1}}{((1 + α_{1}) + (1 - α_{1}) cos 2 θ - 2 β_{1} sin 2 θ)} \end{matrix}

(64)

The eccentricity

\bar{e}

of the ellipse is

{\bar{e}}^{2} = \frac{2 ((1 - α_{1}) cos 2 θ - 2 β_{1} sin 2 θ)}{((1 + α_{1}) + (1 - α_{1}) cos 2 θ - 2 β_{1} sin 2 θ)}

(65)

While the periodic of motion T = 2π/s, where s is given by the relations in Eqs. (58). Finally we demonstrate that the motion of the infinitesimal body around the triangular point will be elliptical and it is given by Eqs. (62) in normal coordinates, where the parameter of motion are given in Eqs. (64, 65).

Conclusion

We conducted a comprehensive analytical study on the effect of the gravitational force of the asteroids belt within frame of the restricted three–body problem. We have formulated the equations of motion of the restricted three–body problem, in the event of perturbation of the asteroids belt. Hence we conducted an analytical study to determine the locations of liberation points and study the linear stability of motion around these points. Furthermore we identified the elements of the periodic orbits of the infinitesimal body in the presence of the asteroids belt perturbation.

Acknowledgements

The Fifth authors (EIA) is partially supported by Fundación Séneca (Spain), grant 20783/PI/18, and Ministry of Science, Innovation and Universities (Spain), grant PGC2018 - 097198 - B -100

References

[1]↑
Abouelmagd, E. I., Alhothuali, M. S., Guirao, L. G. J., Malaikah, H. M. (2015), The effect of zonal harmonic coefficients in the frameworkof the restricted three-body problem, Astrophys. Space Sci. 55: 1660 – 1672. Doi:
- Crossref
- Export Citation
[2]↑
Abouelmagd E. I., Alzahrani F., Guiro J. L. G., Hobiny A. (2016), Periodic orbits around the collinear libration points, J. Nonlinear Sci. Appl. (JNSA). 9 (4): 1716 – 1727. Doi:
- Crossref
- Export Citation
[3]↑
Abouelmagd E. I., Asiri H. M., Sharaf M. A. (2013), The effect of oblateness in the perturbed restricted three-body problem, Meccanica 48: 2479 – 2490. Doi
- Crossref
- Export Citation
[4]↑
Abouelmagd, E.I., Awad, M.E., Elzayat, E.M.A., Abbas, I.A. (2014), Reduction the secular solution to periodic solution in the generalized restricted three-body problem, Astrophys. Space Sci. 341: 495 – 505. Doi:
- Crossref
- Export Citation
[5]↑
Abouelmagd E. I., El-Shaboury S. M. (2012), Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodies, Astrophys. Space Sci. 341: 331 – 341. Doi :
- Crossref
- Export Citation
[6]↑
Abouelmagd E. I. (2012), Existence and stability of triangular points in the restricted three-body problem with numerical applications, Astrophys. Space Sci. 342: 45 – 53. Doi
- Crossref
- Export Citation
[7]↑
Abouelmagd E. I., Guirao J. L.G., Llibre J. (2019), Periodic orbits for the perturbed planar circular restricted 3–body problem, Discrete and Continuous Dynamical Systems - Series B (DCDS-B) 24 (3): 1007 – 1020. Doi:
- Crossref
- Export Citation
[8]↑
Abouelmagd E. I., Mostafa A., Guiro J. L. G. (2015), A first order automated Lie transform. International Journal of Bifurcation and Chaos. 25 (14): 1540026. Doi:
- Crossref
- Export Citation
[9]↑
Abouelmagd E. I., Guirao Juan L. G., Mostafa A. (2014), Numerical integration of the restricted thee-body problem with Lie series, Astrophys. Space Sci. 354 (2)P: 369 – 378. Doi
- Crossref
- Export Citation
[10]↑
Abouelmagd E. I. (2013), The effect of photogravitational force and oblateness in the perturbed restricted three-body problem, Astrophys. Space Sci. 346: 51 – 69. Doi:
- Crossref
- Export Citation
[11]↑
Abouelmagd E. I. (2013), Stability of the triangular points under combined effects of radiation and oblateness in the restricted three–body problem, Earth Moon and Planets 110: 143 – 155. Doi:
- Crossref
- Export Citation
[12]↑
Abouelmagd E. I., Sharaf M. A.(2013), The motion around the libration points in the restricted three–body problem with the effect of radiation and oblateness, Astrophys. Space Sci. 344: 321 – 332. Doi:
- Crossref
- Export Citation
[13]↑
Alzahrani F., Abouelmagd E. I., Guirao J. L. G., Hobiny A. (2017), On the libration collinear points in the restricted three–body problem, Open Physics 15 (1): 58 – 67. Doi:
- Crossref
- Export Citation
[14]↑
Beevi A. S., Sharma, R. K. (2012), Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three–body problem, Astrophys. Space Sci. 340: 245 – 261. Doi:
- Crossref
- Export Citation
[15]↑
Elipe A., Lara M.(1997), Periodic orbits in the restricted three body problem with radiation pressure, Celest. Mech. Dyn. Astron. 68: 1 – 11.
- Crossref
- Export Citation
[16]↑
Elshaboury S. M., Abouelmagd E. I., Kalantonis V. S., Perdios E. A. (2016), The planar restricted three-body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic orbits, Astrophys. Space Sci. 361 (9): 315. Doi:
- Crossref
- Export Citation
[17]↑
Ishwar B., Elipe A. (2001), Secular solutions at triangular equilibrium point inthe generalized photogravitational restricted three body problem, Astrophys. Space Sci. 277: 437 – 446. Doi:
- Crossref
- Export Citation
[18]↑
Greaves J. S., Holland W. S., Moriarty-Schieven G., Jenness T., Dent W. R. F., Zuckerman B., Mccarthy C., Webb R. A., Butner H. M., Gear W. K., Walker H. J. (1998), A dust ring around Eridani: analog to the young Solar System, Astrophys. J. 506: 133 – 137. Doi:
- Crossref
- Export Citation
[19]↑
Jayawardhana R., Holland W. S., Greaves J. S., Dent W. R. F., Marcy G. W., Hartmann L. W., Fazio G. G. (2000), Dust in the 55 Cancri planetary system, The Astrophysical Journal 536, 425 – 428. Doi:
- Crossref
- Export Citation
[20]↑
Jiang I. G., Yeh L. C (2003), Bifurcation for dynamical systems of planet-belt interaction, Int. J. Bifurc. Chaos 13(3): 617 – 630. Doi:.
- Crossref
- Export Citation
[21]↑
Jiang, I.G., Yeh, L.C. (2004b), On the chaotic orbits of disk-star-planet systems, Astron. J. 128: 923 – 932. Doi:.
- Crossref
- Export Citation
[22]↑
Jiang I. G., Yeh L. C. (2004a), The drag-induced resonant capture for Kuiper belt objects, Mon. Not. R. Astron. Soc. 355: 29 – 32. Doi:.
- Crossref
- Export Citation
[23]↑
Jiang Y., Baoyin H. (2019), Periodic orbits related to the equilibrium points in the potential of Irregular–shaped minor celestial bodies, Results in Physics 12: 368 – 374. Doi.org/10.1016/j.rinp.2018.11.049.
- Crossref
- Export Citation
[24]↑
Jung C., Zotos E. E. (2015), Order and chaos in a three dimensional galaxy model, Mechanics Research Communications 69: 45 – 53 Doi.org/10.1016/j.mechrescom.2015.06.005
- Crossref
- Export Citation
[25]↑
Kushvah B.S. (2008), The effect of radiation pressure on the equilibrium points in the generalised photogravitational restricted three body problem, Astrophys. Space Sci. 315: 231–241. Doi:
- Crossref
- Export Citation
[26]↑
Miyamoto M., Nagai R. (1975), Three-dimensional models for the distribution of mass in galaxies, Publ. Astron. Soc. Jpn. 27: 533 – 543.
[27]↑
Papadakis K.E. (2008), Families of asymmetric periodic orbits in the restricted three-body problem, Earth, Moon and Planets, 103: 25 – 42. Doi:
- Crossref
- Export Citation
[28]↑
Pathak N., Abouelmagd E. I., Thomas V. O. (2019), On higher order of resonant periodic orbits in the photogravitational restricted three body problem, J of Astronaut Sci 66(4): 475 – 505. DOI:
- Crossref
- Export Citation
[29]↑
Pathak N., Thomas V. O., Abouelmagd E. I. (2019), The photogravitational restricted three-body problem: Analysis of resonant periodic orbits, Discrete and Continuous Dynamical Systems - Series S (DCDS-S) 12 (4&5): 849 – 875. Doi:
- Crossref
- Export Citation
[30]↑
Pitjeva E. V., Pitjev N. P. (2016), Masses of asteroids and total mass of the main asteroid belt, International Astronomical Union, 318: 212 – 217, Doi:
- Crossref
- Export Citation
[31]↑
Selim H. H., Guirao J. L. G., Abouelmagd E. I. (2019), Libration points in the restricted three–body problem: Euler angles, existence and stability, Discrete and Continuous Dynamical Systems - Series S (DCDS-S) 12 (4&5): 703 – 710. Doi:
- Crossref
- Export Citation
[32]↑
Shibayama M., Yagasaki K.(2011) Families of symmetric relative periodic orbits Originating from the circular Euler solution in the isosceles three–body Problem, Celest. Mech. Dyn. Astron. 110: 53 – 70. Doi:
- Crossref
- Export Citation
[33]↑
Singh J., Begha J. M.(2011), Periodic orbits in the generalized perturbed restricted three-body problem, Astrophys. Space Sci. 332: 319 – 324. Doi:
- Crossref
- Export Citation
[34]↑
Singh J., Taura J.J. (2013), Motion in the generalized restricted three-body problem, Astrophys. Space Sci. 343(1): 95 – 106. Doi:.
- Crossref
- Export Citation
[35]↑
Szebehely V. (1967), Theory of Orbits: The Restricted Problem of Three BodiesAcademic Press, New York
[36]↑
Yeh L. C., Jiang I. G. (2006), On the Chermnykh-like problems: II. The equilibrium points, Astrophys. Space Sci. 306: 189 – 200. Doi:.
- Crossref
- Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

ProCite

RefWorks

Reference Manager

BibTeX

Zotero

EndNote

Article information

Received:: 2019-07-05

Accepted:: 2019-11-20

Published Online:: 2020-08-20

Citation Information:: Applied Mathematics and Nonlinear Sciences, Volume 5, Issue 2, Pages 157–176, eISSN 2444-8656, DOI: https://doi.org/10.2478/amns.2020.2.00022.

	All Time	Past Year	Past 30 Days
Full Text	255	255	41
PDF	110	110	17

Periodic orbit in the frame work of restricted three bodies under the asteroids belt effect

Abstract

1 Introduction

2 Background on asteroids belt potential

3 Model description

4 Locations of equilibrium points

4.1 Location of collinear points

4.1.1 Location of L₁

4.1.2 Location of L₂

4.1.3 Location of L₃

4.2 Location of triangular points

5 Stability of motion around the libration point

5.1 Stability of collinear points

5.2 Stability of triangular points

5.3 Critical mass

6 Periodic orbits

6.1 Periodic orbits around collinear points

6.2 Periodic orbits around triangular points

6.2.1 Elliptic orbits

6.2.2 The orientation of principal axes of the ellipse

Conclusion

References

Article information

Journal + Issues

Applied Mathematics and Nonlinear Sciences

Search

Periodic orbit in the frame work of restricted three bodies under the asteroids belt effect

Abstract

1 Introduction

2 Background on asteroids belt potential

3 Model description

4 Locations of equilibrium points

4.1 Location of collinear points

4.1.1 Location of L1

4.1.2 Location of L2

4.1.3 Location of L3

4.2 Location of triangular points

5 Stability of motion around the libration point

5.1 Stability of collinear points

5.2 Stability of triangular points

5.3 Critical mass

6 Periodic orbits

6.1 Periodic orbits around collinear points

6.2 Periodic orbits around triangular points

6.2.1 Elliptic orbits

6.2.2 The orientation of principal axes of the ellipse

Conclusion

References

Article information

Journal + Issues

Applied Mathematics and Nonlinear Sciences

Journal Information

Search

Sections

Figures

4.1.1 Location of L₁

4.1.2 Location of L₂

4.1.3 Location of L₃