1. bookVolume 5 (2020): Issue 2 (July 2020)
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Copyright
© 2020 Sciendo

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

Published Online: 20 Aug 2020
Page range: 157 - 176
Received: 05 Jul 2019
Accepted: 20 Nov 2019
Journal Details
License
Format
Journal
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Copyright
© 2020 Sciendo
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

MSC 2010

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 (Ln1 and Ln2) 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.

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].

Fig. 1

The asteroids of the inner Solar System and Jupiter (Color figure online)

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 Vb(r,z)=Mb(r2+(a+z2+b2)2)1/2{V_b}\left({r,z} \right) = {{{M_b}} \over {{{\left({{r^2} + {{\left({a + \sqrt {{z^2} + {b^2}} } \right)}^2}} \right)}^{1/2}}}} where

Mb 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 ab(r, z) = ∇Vb (r, z), thereby the acceleration can be written in the following form ab(r,z)=Mb(r2+(a+z2+b2))3/2(x,y,z[1+az2+b2]){{\bf{a}}_b}\left({r,z} \right) = - {{{M_b}} \over {{{\left({{r^2} + \left({a + \sqrt {{z^2} + {b^2}} } \right)} \right)}^{3/2}}}}\left({x,y,z\left[ {1 + {a \over {\sqrt {{z^2} + {b^2}} }}} \right]} \right) If a = b = 0 the potential reduces to the one by a point mass or spherical subject whose mass is Mb. Restricting ourselves to the XY–plane (z = 0), and define Ta + b from Eq. (1) we have Vb(r,0)=Mb(r2+T2)1/2{V_b}(r,0) = {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{1/2}}}} 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 ab(r,0)=Mbr(r2+T2)3/2(x,y){{\bf{a}}_b}\left({r,0} \right) = - {{{M_b}r} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}}\left({x,y} \right)

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 x¨2ny˙=n2x(1μ)r13(xμ)μr23(xμ+1)y¨+2nx˙=n2y(1μ)yr13μyr23\matrix{ {\ddot x - 2n\dot y = {n^2}x - {{\left({1 - \mu } \right)} \over {r_1^3}}\left({x - \mu } \right) - {\mu \over {r_2^3}}\left({x - \mu + 1} \right)} \hfill \cr {\ddot y + 2n\dot x = {n^2}y - {{\left({1 - \mu } \right)y} \over {r_1^3}} - {{\mu y} \over {r_2^3}}} \hfill \cr } In Eq. (5), the angular velocity n = 1, see for details [35], where r12=(xμ)2+y2r22=(xμ+1)2+y2\matrix{ {r_1^2 = (x - \mu {)^2} + {y^2}} \hfill \cr {r_2^2 = (x - \mu + {{1)}^2} + {y^2}} \hfill \cr }

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 x¨2ny˙=Ωxy¨+2nx˙=Ωy\matrix{ {\ddot x - 2n\dot y = {\Omega _x}} \hfill \cr {\ddot y + 2n\dot x = {\Omega _y}} \hfill \cr } where Ω=n2(x2+y2)2+1μr1+μr2+Mb(r2+T2)1/2\Omega = {{{n^2}\left({{x^2} + {y^2}} \right)} \over 2} + {{1 - \mu } \over {{r_1}}} + {\mu \over {{r_2}}} + {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{1/2}}}} and r1 and r2 are given by Eq. (6), while the perturbed mean motion n is n2=1+2Mbrc(rc2+T2)3/2{n^2} = 1 + {{2{M_b}{r_c}} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} here in Eq. (9), rc2=1μ+μ2r_c^2 = 1 - \mu + {\mu ^2} , see [34] for details.

Using Eqs. (7, 8) then the Jacobian integral is governed by x˙2+y˙22Ω(x,y)+C=0{\dot x^2} + {\dot y^2} - 2\Omega \left({x,y} \right) + C = 0 In which C is the constant of integration.

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 x¨=y¨=x˙=y˙=0\ddot x = \ddot y = \dot x = \dot y = 0 , therefore Ωx = Ωy = 0, then we have to solve the following two equations x[n2(1μ)r13μr23Mb(r2+T2)3/2]+μ(1μ)[1r131r23]=0,[n2(1μ)r13μr23Mb(r2+T2)3/2]y=0.\matrix{ \hfill {x\left[ {{n^2} - {{\left({1 - \mu } \right)} \over {r_1^3}} - {\mu \over {r_2^3}} - {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}}} \right] + \mu \left({1 - \mu } \right)\left[ {{1 \over {r_1^3}} - {1 \over {r_2^3}}} \right] = 0,} \cr \hfill {\left[ {{n^2} - {{\left({1 - \mu } \right)} \over {r_1^3}} - {\mu \over {r_2^3}} - {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}}} \right]y = 0.} \cr } Using the above two equations, we will investigate two cases for the locations of equilibrium points in the following subsection.

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 n2x(1μ)(xμ)|xμ|3μ(xμ+1)|xμ+1|3Mbx(r2+T2)3/2=0{n^2}x - {{\left({1 - \mu } \right)\left({x - \mu } \right)} \over {{{\left| {x - \mu } \right|}^3}}} - {{\mu \left({x - \mu + 1} \right)} \over {{{\left| {x - \mu + 1} \right|}^3}}} - {{{M_b}x} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}} = 0

Fig. 2

Configuration of equilibrium points (Color figure online)

Location of L1

The equilibrium point L1 lies beyond masse m1 as in Fig. 2. Then in this case, we have r1r2 = 1 therefor ∂r1/∂x = ∂r2/∂x = −1 and r2 = μx − 1, r1 = μx using Eq. (11), we get n2x+(1μ)(xμ)2+μ(xμ+1)2Mbx(x2+T2)3/2=0{n^2}x + {{\left({1 - \mu } \right)} \over {{{\left({x - \mu } \right)}^2}}} + {\mu \over {{{\left({x - \mu + 1} \right)}^2}}} - {{{M_b}x} \over {{{\left({{x^2} + {T^2}} \right)}^{3/2}}}} = 0 Let r2 = ξ1, r1 = 1 + ξ1, x1 = μ − 1 − ξ1, and n2 = s, then with a help of Eq. (12), we get s(μ1ξ1)+(1μ)(1+ξ1)2+μξ12Mb(μ1ξ1)2(132T2(μ1ξ1)2)=0\matrix{ {s\left({\mu - 1 - {\xi _1}} \right) + {{\left({1 - \mu } \right)} \over {{{\left({1 + {\xi _1}} \right)}^2}}} + {\mu \over {\xi _1^2}} - {{{M_b}} \over {{{\left({\mu - 1 - {\xi _1}} \right)}^2}}}\left({1 - {3 \over 2}{{{T^2}} \over {{{\left({\mu - 1 - {\xi _1}} \right)}^2}}}} \right) = 0} \hfill \cr } after writing Eq. (13) in the series form, we get μ+6μξ1+((1sMb+3T2Mb2)+(10+5s+2Mb)μ)ξ12+((47s4Mb+3T2Mb)+(4+30s+6Mb)μ)ξ13+((621s6Mb+3T2Mb2)+(3+75s+6Mb)μ)ξ14+((435s4Mb)+(2+100s+2Mb)μ)ξ15+((135sMb)+75sμ)ξ16+(21s+30sμ)ξ17+(7s+5sμ)ξ18sξ19=0\matrix{ {\matrix{ {\mu + 6\mu {\xi _1} + \left({\left({1 - s - {M_b} + {{3{T^2}{M_b}} \over 2}} \right) + \left({10 + 5s + 2{M_b}} \right)\mu } \right)\xi _1^2} \hfill \cr { + \left({\left({4 - 7s - 4{M_b} + 3{T^2}{M_b}} \right) + \left({4 + 30s + 6{M_b}} \right)\mu } \right)\xi _1^3} \hfill \cr { + \left({\left({6 - 21s - 6{M_b} + {{3{T^2}{M_b}} \over 2}} \right) + \left({ - 3 + 75s + 6{M_b}} \right)\mu } \right)\xi _1^4} \hfill \cr { + \left({\left({4 - 35s - 4{M_b}} \right) + \left({ - 2 + 100s + 2{M_b}} \right)\mu } \right)\xi _1^5} \hfill \cr { + \left({\left({1 - 35s - {M_b}} \right) + 75s\mu } \right)\xi _1^6 + \left({ - 21s + 30s\mu } \right)\xi _1^7} \hfill \cr { + \left({ - 7s + 5s\mu } \right)\xi _1^8 - s\xi _1^9 = 0} \hfill \cr } } \hfill \cr } Then the parameter μ as a function in the variation ξ1 is given by μ=a11ξ12+a12ξ13+a13ξ14+O[ξ1]5\mu = {a_{11}}\xi _1^2 + {a_{12}}\xi _1^3 + {a_{13}}\xi _1^4 + O{\left[ {{\xi _1}} \right]^5} where a11=1+s+Mb3T2Mb2a12=2+s2Mb+6T2Mba13=8+10s5s2+10Mb7sMb45T2Mb2+152sT2Mb\matrix{ {{a_{11}} = - 1 + s + {M_b} - {{3{T^2}{M_b}} \over 2}} \hfill \cr {{a_{12}} = 2 + s - 2{M_b} + 6{T^2}{M_b}} \hfill \cr {{a_{13}} = - 8 + 10s - 5{s^2} + 10{M_b} - 7s{M_b} - {{45{T^2}{M_b}} \over 2} + {{15} \over 2}s{T^2}{M_b}} \hfill \cr } 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 ξ1=c11μ+c12μ+c13μ3/2{\xi _1} = {c_{11}}\sqrt \mu + {c_{12}}\mu + {c_{13}}{\mu ^{3/2}} where c11=11+s+Mb3T2Mb2c12=(2s+2Mb6T2Mb)(2+2s+2Mb3T2Mb)(1+s+Mb3T2Mb2)c13=(12(2+2s+2Mb3T2Mb)2(11+s+Mb3T2Mb2)3/2×(12+92s55s2+20s3+32Mb128sMb+48s2Mb18T2Mb+240sT2Mb60s2T2Mb)\matrix{ {\matrix{ {{c_{11}} = \sqrt {{1 \over { - 1 + s + {M_b} - {{3{T^2}{M_b}} \over 2}}}} } \hfill \cr {{c_{12}} = {{\left({ - 2 - s + 2{M_b} - 6{T^2}{M_b}} \right)} \over {\left({ - 2 + 2s + 2{M_b} - 3{T^2}{M_b}} \right)\left({ - 1 + s + {M_b} - {{3{T^2}{M_b}} \over 2}} \right)}}} \hfill \cr {{c_{13}} = ({1 \over {2{{\left({ - 2 + 2s + 2{M_b} - 3{T^2}{M_b}} \right)}^2}}}{{\left({{1 \over { - 1 + s + {M_b} - {{3{T^2}{M_b}} \over 2}}}} \right)}^{3/2}}} \hfill \cr {\quad \times \left({\matrix{ { - 12 + 92s - 55{s^2} + 20{s^3} + 32{M_b} - 128s{M_b}} \hfill \cr { + 48{s^2}{M_b} - 18{T^2}{M_b} + 240s{T^2}{M_b} - 60{s^2}{T^2}{M_b}} \hfill \cr } } \right)} \hfill \cr } } \hfill \cr } hence x1=μ1c11μc12μc13μ3/2{x_1} = \mu - 1 - {c_{11}}\sqrt \mu - {c_{12}}\mu - {c_{13}}{\mu ^{3/2}}

Location of L2

Since the point L2 lies between the two primaries, thereby r1 + r2 = 1, r2 = xμ + 1, r1 = μx and ∂r2/∂x = −∂r1/∂x = 1, then by using Eq. (11), we get n2x+(1μ)(xμ)2μ(xμ+1)2Mbx(x2+T2)3/2=0{n^2}x + {{\left({1 - \mu } \right)} \over {{{\left({x - \mu } \right)}^2}}} - {\mu \over {{{\left({x - \mu + 1} \right)}^2}}} - {{{M_b}x} \over {{{\left({{x^2} + {T^2}} \right)}^{3/2}}}} = 0 If we take r2 = ξ2, r1 = 1 − ξ2 and x2 = μ − 1 + ξ2, then we can rewrite Eq. (17) in the following form s(μ1+ξ2)(1μ)(1ξ2)2μξ22Mb(μ1+ξ2)2(132T2(μ1+ξ2)2)=0s\left({\mu - 1 + {\xi _2}} \right) - {{\left({1 - \mu } \right)} \over {{{\left({1 - {\xi _2}} \right)}^2}}} - {\mu \over {\xi _2^2}} - {{{M_b}} \over {{{\left({\mu - 1 + {\xi _2}} \right)}^2}}}\left({1 - {3 \over 2}{{{T^2}} \over {{{\left({\mu - 1 + {\xi _2}} \right)}^2}}}} \right) = 0 again Eq. (18) in the series form is μ+6μξ2+((1Mbs+3MbT2/2)+(20+2Mb+5s)μ)ξ22+((4+4Mb+7s3MbT2)+(366Mb30s)μ)ξ23+((66Mb21s+3MbT2/2)+(33+6Mb+75s)μ)ξ24+((4+4Mb+35s)+(142Mb100s)μ)ξ25+((1Mb35s)+(2+75s)μ)ξ26+(21s30sμ)ξ27+(7s+5sμ)ξ28+sξ29=0\matrix{ {\matrix{ { - \mu + 6\mu {\kern 1pt} {\xi _2} + \left({\left({1 - {M_b} - s + 3{M_b}{T^2}/2} \right) + \left({ - 20 + 2{M_b} + 5s} \right)\mu } \right)\xi _2^2} \hfill \cr { + \left({\left({ - 4 + 4{M_b} + 7s - 3{M_b}{T^2}} \right) + \left({36 - 6{M_b} - 30s} \right)\mu } \right)\xi _2^3} \hfill \cr { + \left({\left({6 - 6{M_b} - 21s + 3{M_b}{T^2}/2} \right) + \left({ - 33 + 6{M_b} + 75s} \right)\mu } \right)\xi _2^4} \hfill \cr { + \left({\left({ - 4 + 4{M_b} + 35s} \right) + \left({14 - 2{M_b} - 100s} \right)\mu } \right)\xi _2^5} \hfill \cr { + \left({\left({1 - {M_b} - 35s} \right) + \left({ - 2 + 75s} \right)\mu } \right)\xi _2^6 + \left({21s - 30s\mu } \right)\xi _2^7} \hfill \cr { + \left({ - 7s + 5s\mu } \right)\xi _2^8 + s\xi _2^9 = 0} \hfill \cr } } \hfill \cr } Then the mass ratio μ as function in ξ2 is given by μ=a11ξ22+a12ξ23+a23ξ24+Oξ25\mu = - {a_{11}}\xi _2^2 + {a_{12}}\xi _2^3 + {a_{23}}\xi _2^4 + O\xi _2^5 where a11 and a12 given by Eq. (14) while a23 is given by a23=2+2ss24Mb+sMb+39T2Mb292sT2Mb{a_{23}} = 2 + 2s - {s^2} - 4{M_b} + s{M_b} + {{39{T^2}{M_b}} \over 2} - {9 \over 2}s{T^2}{M_b} again using the Lagrangian inversion method to inverting the above series, we get ξ2=c11μ+c12μ+c23μ3/2+O[μ]2{\xi _2} = - {c_{11}}\sqrt \mu + {c_{12}}\mu + {c_{23}}{\mu ^{3/2}} + O{\left[ \mu \right]^2} where c11 and c12 given as Eq. (15) while c23 is given by c23=(12(2+2s+2Mb3T2Mb)2)(11sMb+3T2Mb2)3/2×(12+20s+17s24s316Mb32sMb+30T2Mb+144sT2Mb12s2T2Mb)μ3/2)\matrix{ {{c_{23}} = \left({{1 \over {2{{\left({ - 2 + 2s + 2{M_b} - 3{T^2}{M_b}} \right)}^2}}}} \right){{\left({{1 \over {1 - s - {M_b} + {{3{T^2}{M_b}} \over 2}}}} \right)}^{3/2}}} \hfill \cr {\quad \times \left({\matrix{ {12 + 20s + 17{s^2} - 4{s^3} - 16{M_b} - 32s{M_b}} \hfill \cr {\quad + 30{T^2}{M_b} + 144s{T^2}{M_b} - 12{s^2}{T^2}{M_b}} \hfill \cr } } \right){\mu ^{3/2}})} \hfill \cr } hence x2=μ1+c21μ+c22μ+c23μ3/2+O[μ]2{x_2} = \mu - 1 + {c_{21}}\sqrt \mu + {c_{22}}\mu + {c_{23}}{\mu ^{3/2}} + O{\left[ \mu \right]^2}

Location of L3

The point L3 lies beyond the large mass according to Fig. 2, r2r1 = 1, r1 = xμr2 = xμ + 1 and ∂r1/∂x = ∂r2/∂x = 1, then Eq. (11) can be rewritten in the form n2x(1μ)(xμ)2μ(xμ+1)2Mbx(x2+T2)3/2=0{n^2}x - {{\left({1 - \mu } \right)} \over {{{\left({x - \mu } \right)}^2}}} - {\mu \over {{{\left({x - \mu + 1} \right)}^2}}} - {{{M_b}x} \over {{{\left({{x^2} + {T^2}} \right)}^{3/2}}}} = 0 after substituting r2 = ξ3 + 2 and r1 = ξ3 + 1, into Eq. (20), we get s(μ+ξ3+1)(1μ)(1+ξ3)2μ(2+ξ3)2Mb(μ+ξ3+1)2(132T2(μ+ξ3+1)2)=0s\left({\mu + {\xi _3} + 1} \right) - {{\left({1 - \mu } \right)} \over {{{\left({1 + {\xi _3}} \right)}^2}}} - {\mu \over {{{\left({2 + {\xi _3}} \right)}^2}}} - {{{M_b}} \over {{{\left({\mu + {\xi _3} + 1} \right)}^2}}}\left({1 - {3 \over 2}{{{T^2}} \over {{{\left({\mu + {\xi _3} + 1} \right)}^2}}}} \right) = 0 Again the form series of Eq. (21) is given by a30+d30μ+(a31+d31μ)ξ+(a32+d32μ)ξ32+(a33+d33μ)ξ33+(a34+d34μ)ξ34+(a35+d35μ)ξ35+(a36+d36μ)ξ36+(a37+d37μ)ξ37+(a38+d38μ)ξ38+(a39+d39μ)ξ39=0\matrix{ {{a_{30}} + {d_{30}}\mu + \left({{a_{31}} + {d_{31}}\mu } \right)\xi + \left({{a_{32}} + {d_{32}}\mu } \right)\xi _3^2 + \left({{a_{33}} + {d_{33}}\mu } \right)\xi _3^3} \hfill \cr { + \left({{a_{34}} + {d_{34}}\mu } \right)\xi _3^4 + \left({{a_{35}} + {d_{35}}\mu } \right)\xi _3^5 + \left({{a_{36}} + {d_{36}}\mu } \right)\xi _3^6} \hfill \cr { + \left({{a_{37}} + {d_{37}}\mu } \right)\xi _3^7 + \left({{a_{38}} + {d_{38}}\mu } \right)\xi _3^8 + \left({{a_{39}} + {d_{39}}\mu } \right)\xi _3^9 = 0} \hfill \cr } where a30=1Mb+s+3MbT22,d30=34+2Mb+s6MbT2a31=2+2Mb+s6MbT2,d31=746Mb+30MbT2a32=33Mb+15MbT2,d32=4516+12Mb90MbT2a33=(4+4Mb30MbT2,d33=31820Mb+210MbT2a34=55Mb+105MbT22,d34=31564+30Mb420MbT2a35=6+6Mb84MbT2,d35=3816442Mb+756MbT2a36=77Mb+126MbT2,d36=1785256+56Mb1260MT2a37=8+8Mb180MbT2,d37=5116472Mb+1980MbT2a38=99Mb+495MbT22,d38=92071024+90Mb2970MbT2a39=10+10M330MbT2,d39=102351024110Mb+4290MbT2\matrix{{{a_{30}} = - 1 - {M_b} + s + {{3{M_b}{T^2}} \over 2}{\kern 1pt} ,} \hfill & {{d_{30}} = {3 \over 4} + 2{M_b} + s - 6{M_b}{T^2}} \hfill \cr {{a_{31}} = 2 + 2{M_b} + s - 6{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{31}} = - {7 \over 4} - 6{M_b} + 30{M_b}{T^2}} \hfill \cr {{a_{32}} = - 3 - 3{M_b} + 15{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{32}} = {{45} \over {16}} + 12{M_b} - 90{M_b}{T^2}} \hfill \cr {{a_{33}} = (4 + 4{M_b} - 30{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{33}} = \; - {{31} \over 8} - 20{M_b} + 210{M_b}{T^2}} \hfill \cr {{a_{34}} = - 5 - 5{M_b} + {{105{M_b}{T^2}} \over 2}{\kern 1pt} ,} \hfill & {{d_{34}} = {{315} \over {64}} + 30{M_b} - 420{M_b}{T^2}} \hfill \cr {{a_{35}} = 6 + 6{M_b} - 84{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{35}} = - {{381} \over {64}} - 42{M_b} + 756{M_b}{T^2}} \hfill \cr {{a_{36}} = - 7 - 7{M_b} + 126{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{36}} = {{1785} \over {256}} + 56{M_b} - 1260M{T^2}} \hfill \cr {{a_{37}} = 8 + 8{M_b} - 180{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{37}} = - {{511} \over {64}} - 72{M_b} + 1980{M_b}{T^2}} \hfill \cr {{a_{38}} = - 9 - 9{M_b} + {{495{M_b}{T^2}} \over 2}{\kern 1pt} ,} \hfill & {{d_{38}} = {{9207} \over {1024}} + 90{M_b} - 2970{M_b}{T^2}} \hfill \cr {{a_{39}} = 10 + 10M - 330{M_b}{T^2}{\kern 1pt} ,} \hfill & {{d_{39}} = - {{10235} \over {1024}} - 110{M_b} + 4290{M_b}{T^2}} \hfill \cr } Then μ as a function series in ξ3 is given by μ=b30+b31ξ3+b32ξ32+O[ξ33]\matrix{ {\matrix{ {\mu = {b_{30}} + {b_{31}}{\xi _3} + {b_{32}}\xi _3^2 + O[\xi _3^3]} \hfill \cr } } \hfill \cr } where b30=32(22Mb+2s+3MT2)48128Mb64s+384MbT2b31=2(2+18Mb+16Mb236s80Mbs8s2129MbT2+336MbsT2)(38Mb4s+24MbT2)2b32=12(38Mb4s+24MbT2)3(2+78Mb+522s+3608Mbs520s21152Mbs22097MbT220460MbsT2+9600Mbs2T2)\matrix{ {\matrix{ {{b_{30}} = {{32\left({ - 2 - 2{M_b} + 2s + 3M{T^2}} \right)} \over { - 48 - 128{M_b} - 64s + 384{M_b}{T^2}}}} \hfill \cr {{b_{31}} = {{2\left({2 + 18{M_b} + 16M_b^2 - 36s - 80{M_b}s - 8{s^2} - 129{M_b}{T^2} + 336{M_b}s{T^2}} \right)} \over {{{\left({ - 3 - 8{M_b} - 4s + 24{M_b}{T^2}} \right)}^2}}}} \hfill \cr {{b_{32}} = {1 \over {2{{\left({ - 3 - 8{M_b} - 4s + 24{M_b}{T^2}} \right)}^3}}}\left({\matrix{ { - 2 + 78{M_b} + 522s + 3608{M_b}s - 520{s^2} - 1152{M_b}{s^2}} \hfill \cr {\quad - 2097{M_b}{T^2} - 20460{M_b}s{T^2} + 9600{M_b}{s^2}{T^2}} \hfill \cr } } \right)} \hfill \cr } } \hfill \cr }

Now using the Lagrangian inversion method to inverting the above series, we get ξ3=c31μ+c32{\xi _3} = {c_{31}}\mu + {c_{32}} where c31=(38Mb4s+24MbT2)22(2+18Mb36s80Mbs8s2129MbT2)c32=(38Mb4s+24MbT2)2(32(22Mb+2s+3MbT2)48128Mb64s+384MbT2)2(2+18Mb36s80Mbs8s2129MbT2+336MbsT2)\matrix{ {{c_{31}} = {{{{\left({ - 3 - 8{M_b} - 4s + 24{M_b}{T^2}} \right)}^2}} \over {2\left({2 + 18{M_b} - 36s - 80{M_b}s - 8{s^2} - 129{M_b}{T^2}} \right)}}} \hfill \cr {{c_{32}} = {{{{\left({ - 3 - 8{M_b} - 4s + 24{M_b}{T^2}} \right)}^2}\left({ - {{32\left({ - 2 - 2{M_b} + 2s + 3{M_b}{T^2}} \right)} \over { - 48 - 128{M_b} - 64s + 384{M_b}{T^2}}}} \right)} \over {2\left({2 + 18{M_b} - 36s - 80{M_b}s - 8{s^2} - 129{M_b}{T^2} + 336{M_b}s{T^2}} \right)}}} \hfill \cr } hence x3=1+μ+c31μ+c32{x_3} = 1 + \mu + {c_{31}}\mu + {c_{32}} Finally we can use Eqs. (16, 19, 22) to find the locations of collinear points.

Location of triangular points

In the case of the triangular equilibrium points (L4 and L5) y ≠ 0 and Ωx = 0 = Ωy. Using Eqs. (10), we get n2(1μ)r13μr23Mb(r2+T2)3/2=0μ(1μ)r13μ(1μ)r23=0\matrix{ \hfill {{n^2} - {{\left({1 - \mu } \right)} \over {r_1^3}} - {\mu \over {r_2^3}} - {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}} = 0} \cr \hfill {{{\mu \left({1 - \mu } \right)} \over {r_1^3}} - {{\mu \left({1 - \mu } \right)} \over {r_2^3}} = 0} \cr } From the second equation in Eq. (23), we get r1=r2.{r_1} = {r_2}. If we assume that the effect of the potential from the belt is neglected, i.e., Mb = 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 r1=r2=1+ε{r_1} = {r_2} = 1 + \varepsilon From Eqs. (9, 23, 24), we get 1(1+ε)3+2Mbrc(rc2+T2)3/2Mb(r2+T2)3/2=01 - {\left({1 + \varepsilon } \right)^{ - 3}} + {{2{M_b}{r_c}} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}} = 0 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=Mb(2rc1)3(rc2+T2)3/2\varepsilon = - {{{M_b}\left({2{r_c} - 1} \right)} \over {3{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} From Eqs. (6, 24, 26), we get the locations of the triangular points in the following form x=μ12,y=±32(14Mb(2rc1)9(rc2+T2)3/2)x = \mu - {1 \over 2}{\kern 1pt} ,\quad y = \pm {{\sqrt 3 } \over 2}\left({1 - {{4{M_b}\left({2{r_c} - 1} \right)} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}}} \right)

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 x=x0+ξy=y0+η\matrix{ {x = {x_0} + \xi } \hfill \cr {y = {y_0} + \eta } \hfill \cr } We linearize the equations of motion by using Taylor series around equilibrium. Then using Eqs. (7) and Eqs. (28), we obtain ξ¨2nη˙=Ωx0+11!(ξx+ηy)Ωx0+12!(ξx+ηy)2Ωx0+O(3)η¨2nξ˙=Ωy0+11!(ξx+ηy)Ωy0+12!(ξx+ηy)2Ωy0+O(3)\matrix{ {\ddot \xi - 2n\dot \eta = \Omega _x^0 + {1 \over {1!}}\left({\xi {\partial \over {\partial x}} + \eta {\partial \over {\partial y}}} \right)\Omega _x^0 + {1 \over {2!}}{{\left({\xi {\partial \over {\partial x}} + \eta {\partial \over {\partial y}}} \right)}^2}\Omega _x^0 + {\rm{O}}\left(3 \right)} \hfill \cr {\ddot \eta - 2n{\kern 1pt} \dot \xi = \Omega _y^0 + {1 \over {1!}}\left({\xi {\partial \over {\partial x}} + \eta {\partial \over {\partial y}}} \right)\Omega _y^0 + {1 \over {2!}}{{\left({\xi {\partial \over {\partial x}} + \eta {\partial \over {\partial y}}} \right)}^2}\Omega _y^0 + {\rm{O}}\left(3 \right)} \hfill \cr } From the above equations the linear variational equations are ξ¨2nη˙=ξΩxx0+Ωxy0ηη¨+2nξ˙=ξΩxy0+Ωyy0η\matrix{ {\ddot \xi - 2n\dot \eta = \xi \Omega _{xx}^0 + \Omega _{xy}^0\eta } \hfill \cr {\ddot \eta + 2n\dot \xi = \xi \Omega _{xy}^0 + \Omega _{yy}^0\eta } \hfill \cr } 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 Ωxx=n2+3(1μ)(xμ)2r151μr13+3μ(x+1μ)2r25μr23Mb(r2+T2)3/2+3Mbx2(r2+T2)5/2Ωyy=n2+3(1μ)y2r151μr13+3μy2r25μr23Mb(r2+T2)3/2+3Mby2(r2+T2)5/2Ωxy=3(1μ)(xμ)yr15+3μ(x+1μ)yr25+3Mbxy(r2+T2)5/2\matrix{ {{\Omega _{xx}} = {n^2} + {{3\left({1 - \mu } \right){{\left({x - \mu } \right)}^2}} \over {r_1^5}} - {{1 - \mu } \over {r_1^3}} + {{3\mu {{\left({x + 1 - \mu } \right)}^2}} \over {r_2^5}} - {\mu \over {r_2^3}} - {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}} + {{3{M_b}{x^2}} \over {{{\left({{r^2} + {T^2}} \right)}^{5/2}}}}} \hfill \cr {{\Omega _{yy}} = {n^2} + {{3\left({1 - \mu } \right){y^2}} \over {r_1^5}} - {{1 - \mu } \over {r_1^3}} + {{3\mu {y^2}} \over {r_2^5}} - {\mu \over {r_2^3}} - {{{M_b}} \over {{{\left({{r^2} + {T^2}} \right)}^{3/2}}}} + {{3{M_b}{y^2}} \over {{{\left({{r^2} + {T^2}} \right)}^{5/2}}}}} \hfill \cr {{\Omega _{xy}} = {{3\left({1 - \mu } \right)\left({x - \mu } \right)y} \over {{r_1}^5}} + {{3\mu \left({x + 1 - \mu } \right)y} \over {{r_2}^5}} + {{3{M_b}xy} \over {{{\left({{r^2} + {T^2}} \right)}^{5/2}}}}} \hfill \cr } Now we suppose that the solutions of the Eqs. (29) are ξ=Keλt,η=Meλt\xi = K{e^{\lambda t}},\quad \eta = M{e^{\lambda t}} where K, M and λ constant, thereby ξ˙=kλeλt,ξ¨=kλ2eλt,η˙=Mλeλt,η¨=Mλ2eλt.\dot \xi = k\lambda {e^{\lambda t}},\quad \ddot \xi = k{\lambda ^2}{e^{\lambda t}},\quad \dot \eta = M\lambda {e^{\lambda t}},\quad \ddot \eta = M{\lambda ^2}{e^{\lambda t}}. substituting Eqs. (31, 32) into Eqs. (29), then the characteristic equation is determined by |λ2Ωxx2nλΩxy2nλΩxyλ2Ωyy|=0\left| {\matrix{ {{\lambda ^2} - {\Omega _{xx}}} & { - 2n{\kern 1pt} \lambda - {\Omega _{xy}}} \cr {2n{\kern 1pt} \lambda - {\Omega _{xy}}} & {{\lambda ^2} - {\Omega _{yy}}} \cr } } \right| = 0 hence, we get λ4+(4n2Ωxx0Ωyy0)λ2+Ωxx0Ωyy0(Ωxy0)2=0{\lambda ^4} + \left({4{n^2} - \Omega _{xx}^0 - \Omega _{yy}^0} \right){\lambda ^2} + \Omega _{xx}^0\Omega _{yy}^0 - {\left({\Omega _{xy}^0} \right)^2} = 0 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.

Stability of collinear points

At the collinear libration points y = 0, and r1 = |xμ| and r2 = |xμ + 1|. From Eqs. (30, 33), the characteristic equation in case of collinear libration points can be reduced to λ4+(4n2Ωxx0Ωyy0)λ2+Ωxx0Ωyy0=0{\lambda ^4} + \left({4{n^2} - \Omega _{xx}^0 - \Omega _{yy}^0} \right){\lambda ^2} + \Omega _{xx}^0\Omega _{yy}^0 = 0 Let Λ = λ2, Eq. (34) can be rewritten in the form Λ2+bcΛ+cc=0{\Lambda ^2} + {b_c}\Lambda + {c_c} = 0 then Eq. (35) represents a quadratic equation, which its solution can be written in the form Λ1,2=12[bc±bc24cc]{\Lambda _{1,2}} = - {1 \over 2}\left[ {{b_c} \pm \sqrt {b_c^2 - 4{c_c}} } \right] where bc=4n2Ωxx0Ωyy0cc=Ωxx0Ωyy0\matrix{ {{b_c} = } {{\kern 1pt} 4{n^2} - \Omega _{xx}^0 - \Omega _{yy}^0} \hfill \cr {{c_c} = } {{\kern 1pt} \Omega _{xx}^0\Omega _{yy}^0} \hfill \cr } Then with a help of Eqs. (36, 37), the roots of characteristic Eq. (34) can be write as λ1,2 = ±σ, λ3,4 = ± where σ and τ are real number, which can be calculated by σ2=12[bc24ccbc]τ2=12[bc24cc+bc]\matrix{ {{\sigma ^2} = {1 \over 2}\left[ {\sqrt {b_c^2 - 4{c_c}} - {b_c}} \right]} \hfill \cr {{\tau ^2} = {1 \over 2}\left[ {\sqrt {b_c^2 - 4{c_c}} + {b_c}} \right]} \hfill \cr } From the above two equations, we can determine the eigenvalues and the properties of the equilibrium point.

Stability of triangular points

At the triangular points, we have Ωxx0=34(1+5Mb(2rc1)3(rc2+T2)3/2+Mb(μ1/2)23(rc2+T2)5/2)Ωyy0=94(1+7Mb(2rc1)9(rc2+T2)3/2+Mb(rc2+T2)5/2)Ωxy0=334(12μ)(1+Mb(rc2+T2)5/211Mb(2rc1)9(rc2+T2)3/2)\matrix{ {\Omega _{xx}^0 = } {{3 \over 4}\left({1 + {{5{M_b}\left({2{r_c} - 1} \right)} \over {3{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{{M_b}{{\left({\mu - 1/2} \right)}^2}} \over {3{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)} \hfill \cr {\Omega _{yy}^0 = } {{9 \over 4}\left({1 + {{7{M_b}\left({2{r_c} - 1} \right)} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)} \hfill \cr {\Omega _{xy}^0 = } { \mp {{3\sqrt 3 } \over 4}(1 - 2\mu)\left({1 + {{{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}} - {{11{M_b}(2{r_c} - 1)} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}}} \right)} \hfill \cr } and substituting λ2 = ω into the characteristic Eq. (33), then we get ω2+bω+c=0{\omega ^2} + b{\kern 1pt} \omega + c = 0 where a and c can be evaluated with a help of Eqs. (38) and the following two relations b=4n2Ωxx0Ωyy0c=Ωxx0Ωyy0(Ωxy0)2\matrix{ {b = {\kern 1pt} } {4{n^2} - \Omega _{xx}^0 - \Omega _{yy}^0} \hfill \cr {c = {\kern 1pt} } {\Omega _{xx}^0\Omega _{yy}^0 - {{(\Omega _{xy}^0)}^2}} \hfill \cr } hence we have b=1+Mb(2rc+3)(rc2+T2)3/23Mbrc2(rc2+T2)5/2c=(27433Mb(2rc1)2(rc2+T2)3/227Mb4(rc2+T2)5/2)μ2+(274+33Mb(2rc1)2(rc2+T2)3/2+27Mb4(rc2+T2)5/2)μ\matrix{ {b = } {1 + {{{M_b}\left({2{r_c} + 3} \right)} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{3{M_b}r_c^2} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \hfill \cr {c = } {\left({ - {{27} \over 4} - {{33{M_b}\left({2{r_c} - 1} \right)} \over {2{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{27{M_b}} \over {4{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right){\mu ^2}} \hfill \cr {\quad + \left({{{27} \over 4} + {{33{M_b}\left({2{r_c} - 1} \right)} \over {2{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{27{M_b}} \over {4{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)\mu } \hfill \cr } Thereby the roots of Eq. (39) are given by ω1,2=12[bD]{\omega _{1,2}} = - {1 \over 2}\left[ {b \mp \sqrt D } \right] where D = b2 − 4c is discriminant D=(27+66Mb(2rc1)(rc2+T2)3/2+27Mb(rc2+T2)5/2)μ2(27+66Mb(2rc1)(rc2+T2)3/2+27Mb(rc2+T2)5/2)μ+1+2Mb(2rc+3)(rc2+T2)3/26Mbrc2(rc2+T2)5/2\matrix{ {D = \left({27 + {{66{M_b}\left({2{r_c} - 1} \right)} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{27{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right){\mu ^2}} \hfill \cr {\quad - \left({27 + {{66{M_b}\left({2{r_c} - 1} \right)} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{27{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)\mu } \hfill \cr {\quad + 1 + {{2{M_b}\left({2{r_c} + 3} \right)} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{6{M_b}r_c^2} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \hfill \cr } which can be written as of a function of the mass parameter μ in the form D=αμ2βμ+γD = \alpha {\mu ^2} - \beta \mu + \gamma here α and β are the coefficients μ2 and μ respectively. Using Eq. (41) the roots are given by λ1,2=±ω1,λ3,4=±ω2{\lambda _{1,2}} = \pm \sqrt {{\omega _1}} {\kern 1pt} {\kern 1pt} ,\quad {\lambda _{3,4}} = \pm \sqrt {{\omega _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={1+(2Mb(2rc+3)(rc2+T2)3/26Mbrc2(rc2+T2)5/2)>0,whenμ=0(234+Mb(58rc45)2(rc2+T2)3/2+3Mb(8rc2+9)4(rc2+T2)5/2)<0,whenμ=1/2D = \left\{ {\matrix{{1 + \left( {{{2{M_b}\left( {2{r_c} + 3} \right)} \over {{{\left( {r_c^2 + {T^2}} \right)}^{3/2}}}} - {{6{M_b}r_c^2} \over {{{\left( {r_c^2 + {T^2}} \right)}^{5/2}}}}} \right) > 0{\kern 1pt} ,{\rm{when}}{\kern 1pt} {\kern 1pt} \mu = 0} \hfill \cr { - \left( {{{23} \over 4} + {{{M_b}\left( {58{r_c} - 45} \right)} \over {2{{\left( {r_c^2 + {T^2}} \right)}^{3/2}}}} + {{3{M_b}\left( {8r_c^2 + 9} \right)} \over {4{{\left( {r_c^2 + {T^2}} \right)}^{5/2}}}}} \right) < 0{\kern 1pt} ,{\rm{when}}{\kern 1pt} {\kern 1pt} \mu = 1/2} \hfill \cr } } \right. Also the derivative of D with respect to the parameter μ is given by dDdμ=2(27+66Mb(2rc1)(rc2+T2)3/2+27Mb(rc2+T2)5/2)μ(27+66Mb(2rc1)(rc2+T2)3/2+27Mb(rc2+T2)5/2)\matrix{ {{{dD} \over {d\mu }} = 2\left({27 + {{66{M_b}\left({2{r_c} - 1} \right)} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{27{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)\mu } \hfill \cr {\quad\,\,\, - \left({27 + {{66{M_b}\left({2{r_c} - 1} \right)} \over {{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{27{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)} \hfill \cr } thereby dDdμ={27(66Mb(2rc1)(rc2+T2)3/2+27Mb(rc2+T2)5/2)<0,whenμ=00,whenμ=1/2{{dD} \over {d\mu }} = \left( {\matrix{{ - 27 - \left( {{{66{M_b}\left( {2{r_c} - 1} \right)} \over {{{\left( {r_c^2 + {T^2}} \right)}^{3/2}}}} + {{27{M_b}} \over {{{\left( {r_c^2 + {T^2}} \right)}^{5/2}}}}} \right) < 0{\kern 1pt} ,} \hfill & {{\rm{when}}{\kern 1pt} {\kern 1pt} \mu = 0} \hfill \cr {0{\kern 1pt} {\kern 1pt} ,} \hfill & {{\rm{when}}{\kern 1pt} {\kern 1pt} \mu = 1/2} \hfill \cr } } \right. because Mb ≪ 1, and using Eqs. (44, 45), we obtain dDdμ0μ(0,1/2){{dD} \over {d\mu }} \le 0{\kern 1pt} {\kern 1pt} \forall {\kern 1pt} {\kern 1pt} \mu \in \left({0,1/2} \right)

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 (μc). Consequently we will examine three possible cases for the value of μ.

If 0 < μ < μc implies D = b2 −4c > 0, and D decreasing in the interval (0, 1/2). Since b > 0, b24c<b2(b24c<b){b^2} - 4c < {b^2}\left({\sqrt {{b^2} - 4c} < b} \right) 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 = μc (D = 0), then we have double equal roots of λ which lead to secular terms, thereby the triangular points are unstable.

When μc < 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.

Critical mass

Under the previous discussion, when Eq. (43) is equal zero, then one can obtain the value of critical mass (μc), which is governed by μc=12α[α+α24αγ]{\mu _c} = - {1 \over {2\alpha }}\left[ {\alpha + \sqrt {{\alpha ^2} - 4\alpha \gamma } } \right] where α2=729+3564Mb(2rc1)(rc2+T2)3/2+1458Mb(rc2+T2)5/24αγ=108+216Mb(2rc+3)(rc2+T2)3/2648Mbrc2(rc2+T2)5/2+264Mb(2rc1)(rc2+T2)3/2+108Mb(rc2+T2)5/2\matrix{{\;\;{\alpha ^2} = 729 + {{3564{M_b}\left( {2{r_c} - 1} \right)} \over {{{\left( {r_c^2 + {T^2}} \right)}^{3/2}}}} + {{1458{M_b}} \over {{{\left( {r_c^2 + {T^2}} \right)}^{5/2}}}}} \hfill \cr {4\alpha \gamma = 108 + {{216{M_b}\left( {2{r_c} + 3} \right)} \over {{{\left( {r_c^2 + {T^2}} \right)}^{3/2}}}} - {{648{M_b}r_c^2} \over {{{\left( {r_c^2 + {T^2}} \right)}^{5/2}}}}} \hfill \cr {\;\;\;\;\;\;\;\;\; + {{264{M_b}\left( {2{r_c} - 1} \right)} \over {{{\left( {r_c^2 + {T^2}} \right)}^{3/2}}}} + {{108{M_b}} \over {{{\left( {r_c^2 + {T^2}} \right)}^{5/2}}}}} \hfill \cr } utilizing Eqs. (46, 47) we get μc=μ0+μp{\mu _c} = {\mu _0} + {\mu _p}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 (μp) 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 μc = μ0, where the values of μ0 and μp are given by the following relations μ0=12(1699),μp=(768rc)Mb2769(rc2+T2)3/2(1+6rc2)Mb369(rc2+T2)5/2{\mu _0} = {1 \over 2}\left({1 - {{\sqrt {69} } \over 9}} \right){\kern 1pt} ,\quad {\mu _p} = {{\left({76 - 8{r_c}} \right){M_b}} \over {27\sqrt {69} {{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{\left({1 + 6r_c^2} \right){M_b}} \over {3\sqrt {69} {{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}

Periodic orbits
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 Kj and Mj, it is governed by Mj=αjKj{M_j} = {\alpha _j}{K_j} where αj=(λj2Ωxx0)2nλj=2nλjΩyy0λj2{\alpha _j} = {{\left({\lambda _j^2 - \Omega _{xx}^0} \right)} \over {2n{\lambda _j}}} = {{2n{\lambda _j}} \over {\Omega _{yy}^0 - \lambda _j^2}} This relation means that the coefficients Kj and Mj (j = 1, 2, 3, 4) are dependent. Therefore the four initial conditions ξ0, η0, ξ˙0{\dot \xi _0} and η˙0{\dot \eta _0} associated with Eqs. (29) will determine the two sets of coefficients and will completely determine the eight coefficients (Kj, Mj). Where the subscript 0 indicates to these quantities are evaluated at the initial time (t = t0). Substituting Eq. (49) into Eq. (31), we get ξ0=j=14Kjeλjtξ˙0=j=14Kjλjeλjtη0=j=14Kjαjeλjtη˙0=j=14Kjλjαjeλjt\matrix{{{\xi _0} = \sum\limits_{j = 1}^4 {K_j}{e^{{\lambda _j}t}}} \hfill & {{{\dot \xi }_0} = \sum\limits_{j = 1}^4 {K_j}{\lambda _j}{e^{{\lambda _j}t}}} \hfill \cr {{\eta _0} = \sum\limits_{j = 1}^4 {K_j}{\alpha _j}{e^{{\lambda _j}t}}} \hfill & {{{\dot \eta }_0} = \sum\limits_{j = 1}^4 {K_j}{\lambda _j}{\alpha _j}{e^{{\lambda _j}t}}} \hfill \cr } The coefficient can be expressed as function of the initial conditions, because the determinant (Δ) of System of Eqs. (50) is not zero. Δ=Ωxx0Ωyy0[(Ωxx0+Ωyy04n)212Ωxx0Ωyy0]0\Delta = - \sqrt {{{\Omega _{xx}^0} \over {\Omega _{yy}^0}}} \left[ {{{\left({\Omega _{xx}^0 + \Omega _{yy}^0 - 4n} \right)}^2} - {1 \over 2}\Omega _{xx}^0\Omega _{yy}^0} \right] \ne 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 ξ=K1cosσ(tt0)+K2sinσ(tt0)+K3cosτ(tt0)+K4sinτ(tt0),η=M1cosσ(tt0)+M2sinσ(tt0)+M3cosτ(tt0)+M4sinτ(tt0).\matrix{ {\xi = } {{K_1}\cos \sigma (t - {t_0}) + {K_2}\sin \sigma (t - {t_0}) + {K_3}\cos \tau (t - {t_0}) + {K_4}\sin \tau (t - {t_0}),} \hfill \cr {\eta = } {{M_1}\cos \sigma (t - {t_0}) + {M_2}\sin \sigma (t - {t_0}) + {M_3}\cos \tau (t - {t_0}) + {M_4}\sin \tau (t - {t_0}).} \hfill \cr } Now We can take K1 = K2 = 0, therefor the solution in Eqs. (51) can rewritten in the following form ξ=ξ0cosτ(tt0)+η0β3sinτ(tt0)η=η0cosτ(tt0)ξ0β3sinτ(tt0)\matrix{ {\xi = } {{\xi _0}\cos \tau \left({t - {t_0}} \right) + {{{\eta _0}} \over {{\beta _3}}}\sin \tau (t - {t_0})} \hfill \cr {\eta = } {{\eta _0}\cos \tau \left({t - {t_0}} \right) - {\xi _0}{\beta _3}\sin \tau (t - {t_0})} \hfill \cr } where the real quantities τ and β3 are defined by τ={12[(4n2Ωxx0Ωyy0)24Ωxx0Ωyy0+4n2Ωxx0Ωyy0]}1/2β3=τ2+Ωxx02nτ=2nττ2+Ωyy0\matrix{ {\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\tau = } {{\kern 1pt} {{\left\{ {{1 \over 2}\left[ {\sqrt {{{\left({4{n^2} - \Omega _{xx}^0 - \Omega _{yy}^0} \right)}^2} - 4\Omega _{xx}^0\Omega _{yy}^0} + 4{n^2} - \Omega _{xx}^0 - \Omega _{yy}^0} \right]} \right\}}^{1/2}}} \hfill \cr {{\beta _3} = } {{\kern 1pt} {{{\tau ^2} + \Omega _{xx}^0} \over {2n\tau }} = {{2n\tau } \over {{\tau ^2} + \Omega _{yy}^0}}} \hfill \cr } in which λ3 = and α3 = 3

From Eqs. (52) we get the velocity variation in the form ξ˙=ξ0τsinτ(tt0)+η0β3τcosτ(tt0)η˙=η0τsinτ(tt0)ξ0β3τcosτ(tt0)\matrix{ {\dot \xi = } { - {\xi _0}\tau \sin \tau \left({t - {t_0}} \right) + {{{\eta _0}} \over {{\beta _3}}}\tau \cos \tau (t - {t_0})} \hfill \cr {\dot \eta = } { - {\eta _0}\tau \sin \tau \left({t - {t_0}} \right) - {\xi _0}{\beta _3}\tau \cos \tau (t - {t_0})} \hfill \cr } Using Eqs. (53), when t = t0, one obtains ξ˙0=η0τ/β3,η˙0=ξ0β3τ{\dot \xi _0} = {\eta _0}\tau /{\beta _3}{\kern 1pt} {\kern 1pt} ,\quad \quad {\dot \eta _0} = - {\xi _0}{\beta _3}\tau With a help of Eq. (54) we can eliminate the time from Eq. (52), and the equation of periodic orbits reduced to ξ2β32+η2=η02+ξ02β32{\xi ^2}\beta _3^2 + {\eta ^2} = \eta _0^2 + \xi _0^2\beta _3^2 which can be rewritten in the following standard form ξ2(η02+ξ02β32)/β32+η2(η02+ξ02β32)=1{{{\xi ^2}} \over {\left({\eta _0^2 + \xi _0^2\beta _3^2} \right)/\beta _3^2}} + {{{\eta ^2}} \over {\left({\eta _0^2 + \xi _0^2\beta _3^2} \right)}} = 1Eq. (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 (ac), the semi-minor axes (bc) and the eccentricity (ec) are given by ac2=(η02+ξ02β32),ac2=(η02+ξ02β32),ec2=(11β32)a_c^2 = \left({\eta _0^2 + \xi _0^2\beta _3^2} \right){\kern 1pt} ,\quad a_c^2 = \left({\eta _0^2 + \xi _0^2\beta _3^2} \right){\kern 1pt} ,\quad e_c^2 = \left({1 - {1 \over {\beta _3^2}}} \right) where the semi–major axes parallel to the η–axes while the semi-minor axes parallel to the ξ–axes. Also the periodic time (Tc) can be calculated by Tc = 2π/τ. Since η˙0=ξ0β3τ{\dot \eta _0} = - {\xi _0}{\beta _3}\tau , ξ˙0=0{\dot \xi _0} = 0 at ξ0 0 and η0 = 0, the motion along the orbits is retrograde.

Periodic orbits around triangular points

The triangular points are linearly stable in the range 0 < μ < μc. 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 ξ=C1coss1t+D1sins1t+C2coss2t+D2sins2t,η=C¯1coss1t+D¯1sins1t+C¯2coss2t+D¯2sins2t.\matrix{ {\xi = } {{C_1}\cos {s_1}t + {D_1}\sin {s_1}t + {C_2}\cos {s_2}t + {D_2}\sin {s_2}t,} \hfill \cr {\eta = } {{{\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.} \hfill \cr } Therefor the terms with coefficients C1, D1, C¯1{\bar C_1} , D¯1{\bar D_1} associated with angular frequencies s1 (mean motion) refer to the long periodic orbits and terms with coefficients C2, D2, C¯2{\bar C_2} , D¯2{\bar D_2} associated with angular frequencies s2, which refer to the short periodic orbits. In addition s1,22=ωs_{1,2}^2 = - \omega , thereby we get s1,22=12[bD]s_{1,2}^2 = {1 \over 2}\left[ {b \mp \sqrt D } \right] Substituting Eqs. (42, 40) into Eq. (57), with some simple calculations, the angular frequencies of motion s1 and s2 are given by s1=332μ(1μ)(1+11Mb(2rc1)9(rc2+T2)3/2+Mb2(rc2+T2)5/2)s2=1278μ(1μ)(1+22Mb(2rc1)9(rc2+T2)3/2+Mb(rc2+T2)5/2)+Mb(2rc+3)2(rc2+T2)3/23Mbrc22(rc2+T2)5/2\matrix{ {{s_1} = {{3\sqrt 3 } \over 2}\sqrt {\mu \left({1 - \mu } \right)} \left({1 + {{11{M_b}\left({2{r_c} - 1} \right)} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{{M_b}} \over {2{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)} \hfill \cr {\matrix{ {{s_2} = 1 - {{27} \over 8}\mu \left({1 - \mu } \right)\left({1 + {{22{M_b}\left({2{r_c} - 1} \right)} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} + {{{M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right)} \hfill \cr {\quad + {{{M_b}\left({2{r_c} + 3} \right)} \over {2{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{3{M_b}r_c^2} \over {2{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \hfill \cr } } \hfill \cr } 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 C¯i=Γi(2nDisiΩxy0Ci)D¯i=Γi(2nCisi+Ωxy0Di)\matrix{ {{{\bar C}_i} = } {{\Gamma _i}\left({2n{D_i}{s_i} - \Omega _{xy}^0{C_i}} \right)} \hfill \cr {{{\bar D}_i} = } { - {\Gamma _i}\left({2n{C_i}{s_i} + \Omega _{xy}^0{D_i}} \right)} \hfill \cr } where Γi=1si2+Ωyy0=si2+Ωxx04n2si2+(Ωxy0)2>0{\Gamma _i} = {1 \over {s_i^2 + \Omega _{yy}^0}} = {{s_i^2 + \Omega _{xx}^0} \over {4{n^2}s_i^2 + {{(\Omega _{xy}^0)}^2}}} > 0 here (i = 1, 2) and Ωxx0\Omega _{{\rm{xx}}}^0 , Ωxy0\Omega _{{\rm{xy}}}^0 and Ωyy0\Omega _{{\rm{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, ξ˙0{\dot \xi _0} and η˙0{\dot \eta _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. C2=D2=C¯2=D¯2=0{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 ξ0=C1,η0=C¯1,ξ˙0=D1s1,η˙0=D¯1s1{\xi _0} = {C_1}{\kern 1pt} {\kern 1pt} ,\quad {\eta _0} = {\bar C_1}{\kern 1pt} ,\quad {\dot \xi _0} = {D_1}{s_1}{\kern 1pt} {\kern 1pt} ,\quad {\dot \eta _0} = {\bar D_1}{s_1} where D1=ξ0Ωxy0+η0(Ωyy0+s12)2ns1,D¯1=ξ0(Ωxx0+s12)+η0Ωxy02ns1{D_1} = {{{\xi _0}\Omega _{xy}^0 + {\eta _0}\left({\Omega _{yy}^0 + s_1^2} \right)} \over {2n{s_1}}}{\kern 1pt} ,\quad {\bar D_1} = {{{\xi _0}\left({\Omega _{xx}^0 + s_1^2} \right) + {\eta _0}\Omega _{xy}^0} \over {2n{s_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 ξ0=12μ,η0=32(14Mb(2rc1)9(rc2+T2)3/2){\xi _0} = {1 \over 2} - \mu {\kern 1pt} ,\quad {\eta _0} = \mp {{\sqrt 3 } \over 2}\left({1 - {{4{M_b}\left({2{r_c} - 1} \right)} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}}} \right) 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 ξ=C1coss1t+D1sins1t,η=C¯1coss1t+D¯1sins1t.\matrix{ {\xi = } {{C_1}\cos {s_1}t + {D_1}\sin {s_1}t,} \hfill \cr {\eta = } {{{\bar C}_1}\cos {s_1}t + {{\bar D}_1}\sin {s_1}t.} \hfill \cr } After eliminating the time from Eqs. (59), we get α1ξ2+η2+2β1ξη=γ1{\alpha _1}{\xi ^2} + {\eta ^2} + 2{\beta _1}\xi \eta = {\gamma _1} Since |Γ12(4n2s12+(Ωxy0)2)Γ1Ωxy0Γ1Ωxy01|=(2ns1Γ1)2>0\left| {\matrix{ {\Gamma _1^2\left({4{n^2}s_1^2 + {{(\Omega _{xy}^0)}^2}} \right)} & {{\Gamma _1}\Omega _{xy}^0} \cr {{\Gamma _1}\Omega _{xy}^0} & 1 \cr } } \right| = {\left({2n{s_1}{\Gamma _1}} \right)^2} > 0Eq. (60) represents of an ellipse with center at the origin coordinate system (ξ, η), which is coinciding with L4 or L4, where α1=4n2s12+(Ωxy0)2(s12+Ωyy0)2,β1=Ωxy0s12+Ωyy0,γ1=α1ξ02+2β1ξ0η0+η02{\alpha _1} = {{4{n^2}s_1^2 + {{(\Omega _{xy}^0)}^2}} \over {{{\left({s_1^2 + \Omega _{yy}^0} \right)}^2}}}{\kern 1pt} {\kern 1pt} ,\quad {\beta _1} = {{\Omega _{xy}^0} \over {s_1^2 + \Omega _{yy}^0}}{\kern 1pt} {\kern 1pt} ,\quad {\gamma _1} = {\alpha _1}\xi _0^2 + 2{\beta _1}{\xi _0}{\eta _0} + \eta _0^2

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 (ξ¯,η¯)\left({\bar \xi ,\bar \eta } \right) called normal coordinates such that the bilinear term dose not appear. Hence the old and new coordinates system are related by the following equation ξ=ξ¯cosθη¯sinθη=ξ¯sinθ+η¯cosθ\matrix{ {\xi = } {\bar \xi \cos \theta - \bar \eta \sin \theta } \hfill \cr {\eta = } {\bar \xi \sin \theta + \bar \eta \cos \theta } \hfill \cr } Substituting Eqs. (61) into Eq. (60) and equate the coefficient of ξ¯η¯\bar \xi \bar \eta by zero, therefor after simplify the equation, we get ξ¯2a¯2+η¯2b¯2=1{{{{\bar \xi }^2}} \over {{{\bar a}^2}}} + {{{{\bar \eta }^2}} \over {{{\bar b}^2}}} = 1 where the orientation of the principal axes is governed by tan2θ=2Ωxy0Ωxx0Ωyy0\tan 2\theta = {{2\Omega _{xy}^0} \over {\Omega _{xx}^0 - \Omega _{yy}^0}} From Eqs. (38, 63), we obtain tan2θ=±3(12μ+8(12μ)(2rc1)Mb9(rc2+T2)3/22μ(13μ)Mb(rc2+T2)5/2)\tan 2\theta = \pm \sqrt 3 \left({1 - 2\mu + {{8\left({1 - 2\mu } \right)\left({2{r_c} - 1} \right){M_b}} \over {9{{\left({r_c^2 + {T^2}} \right)}^{3/2}}}} - {{2\mu \left({1 - 3\mu } \right){M_b}} \over {{{\left({r_c^2 + {T^2}} \right)}^{5/2}}}}} \right) 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 a¯2=2γ1((1+α1)(1α1)cos2θ+2β1sin2θ)b¯2=2γ1((1+α1)+(1α1)cos2θ2β1sin2θ)\matrix{ {{{\bar a}^2} = } {{{2{\gamma _1}} \over {\left({\left({1 + {\alpha _1}} \right) - \left({1 - {\alpha _1}} \right)\cos 2\theta + 2{\beta _1}\sin 2\theta } \right)}}} \hfill \cr {{{\bar b}^2} = } {{{2{\gamma _1}} \over {\left({\left({1 + {\alpha _1}} \right) + \left({1 - {\alpha _1}} \right)\cos 2\theta - 2{\beta _1}\sin 2\theta } \right)}}} \hfill \cr } The eccentricity e¯\bar e of the ellipse is e¯2=2((1α1)cos2θ2β1sin2θ)((1+α1)+(1α1)cos2θ2β1sin2θ){\bar e^2} = {{2\left({\left({1 - {\alpha _1}} \right)\cos 2\theta - 2{\beta _1}\sin 2\theta } \right)} \over {\left({\left({1 + {\alpha _1}} \right) + \left({1 - {\alpha _1}} \right)\cos 2\theta - 2{\beta _1}\sin 2\theta } \right)}}

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.

Fig. 1

The asteroids of the inner Solar System and Jupiter (Color figure online)
The asteroids of the inner Solar System and Jupiter (Color figure online)

Fig. 2

Configuration of equilibrium points (Color figure online)
Configuration of equilibrium points (Color figure online)

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:10.1016/j.asr.2014.12.030AbouelmagdE. I.AlhothualiM. S.GuiraoL. G. J.MalaikahH. M.2015The effect of zonal harmonic coefficients in the frameworkof the restricted three-body problemAstrophys. Space Sci.551660167210.1016/j.asr.2014.12.030Open DOISearch in Google Scholar

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:10.22436/jnsa.009.04.27AbouelmagdE. I.AlzahraniF.GuiroJ. L. G.HobinyA.2016Periodic orbits around the collinear libration pointsJ. Nonlinear Sci. Appl. (JNSA).941716172710.22436/jnsa.009.04.27Open DOISearch in Google Scholar

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 10.1007/S11012-013-9762-3AbouelmagdE. I.AsiriH. M.SharafM. A.2013The effect of oblateness in the perturbed restricted three-body problemMeccanica482479249010.1007/S11012-013-9762-3Open DOISearch in Google Scholar

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:10.1007/s10509-013-1756-zAbouelmagdE.I.AwadM.E.ElzayatE.M.A.AbbasI.A.2014Reduction the secular solution to periodic solution in the generalized restricted three-body problemAstrophys. Space Sci.34149550510.1007/s10509-013-1756-zOpen DOISearch in Google Scholar

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 : 10.1007/s10509-012-1093-7AbouelmagdE. I.El-ShabouryS. M.2012Periodic orbits under combined effects of oblateness and radiation in the restricted problem of three bodiesAstrophys. Space Sci.34133134110.1007/s10509-012-1093-7Open DOISearch in Google Scholar

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 10.1007/s10509-012-1162-yAbouelmagdE. I.2012Existence and stability of triangular points in the restricted three-body problem with numerical applicationsAstrophys. Space Sci.342455310.1007/s10509-012-1162-yOpen DOISearch in Google Scholar

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:10.3934/dcdsb.2019003AbouelmagdE. I.GuiraoJ. L.G.LlibreJ.2019Periodic orbits for the perturbed planar circular restricted 3–body problemDiscrete and Continuous Dynamical Systems - Series B (DCDS-B)2431007102010.3934/dcdsb.2019003Open DOISearch in Google Scholar

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:10.1142/S021812741540026XAbouelmagdE. I.MostafaA.GuiroJ. L. G.2015A first order automated Lie transformInternational Journal of Bifurcation and Chaos25141540026.10.1142/S021812741540026XOpen DOISearch in Google Scholar

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 10.1007/s10509-014-2107-4AbouelmagdE. I.Guirao JuanL. G.MostafaA.2014Numerical integration of the restricted thee-body problem with Lie seriesAstrophys. Space Sci.354236937810.1007/s10509-014-2107-4Open DOISearch in Google Scholar

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: 10.1007/s10509-013-1439-9AbouelmagdE. I.2013The effect of photogravitational force and oblateness in the perturbed restricted three-body problemAstrophys. Space Sci.346516910.1007/s10509-013-1439-9Open DOISearch in Google Scholar

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: 10.1007/s11038-013-9415-5AbouelmagdE. I.2013Stability of the triangular points under combined effects of radiation and oblateness in the restricted three–body problemEarth Moon and Planets11014315510.1007/s11038-013-9415-5Open DOISearch in Google Scholar

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: 10.1007/s10509-012-1335-8AbouelmagdE. I.SharafM. A.2013The motion around the libration points in the restricted three–body problem with the effect of radiation and oblatenessAstrophys. Space Sci.34432133210.1007/s10509-012-1335-8Open DOISearch in Google Scholar

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:10.1515/phys-2017-0007AlzahraniF.AbouelmagdE. I.GuiraoJ. L. G.HobinyA.2017On the libration collinear points in the restricted three–body problemOpen Physics151586710.1515/phys-2017-0007Open DOISearch in Google Scholar

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: 10.1007/s10509-012-1052-3BeeviA. S.SharmaR. K.2012Oblateness effect of Saturn on periodic orbits in the Saturn-Titan restricted three–body problemAstrophys. Space Sci.34024526110.1007/s10509-012-1052-3Open DOISearch in Google Scholar

Elipe A., Lara M.(1997), Periodic orbits in the restricted three body problem with radiation pressure, Celest. Mech. Dyn. Astron. 68: 1 – 11.ElipeA.LaraM.1997Periodic orbits in the restricted three body problem with radiation pressureCelest. Mech. Dyn. Astron.68111Search in Google Scholar

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:10.1007/s10509-016-2894-xElshabouryS. M.AbouelmagdE. I.KalantonisV. S.PerdiosE. A.2016The planar restricted three-body problem when both primaries are triaxial rigid bodies: Equilibrium points and periodic orbitsAstrophys. Space Sci.361931510.1007/s10509-016-2894-xOpen DOISearch in Google Scholar

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: 10.1023/A:1012528929233IshwarB.ElipeA.2001Secular solutions at triangular equilibrium point inthe generalized photogravitational restricted three body problemAstrophys. Space Sci.27743744610.1023/A:1012528929233Open DOISearch in Google Scholar

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:10.1086/311652GreavesJ. S.HollandW. S.Moriarty-SchievenG.JennessT.DentW. R. F.ZuckermanB.MccarthyC.WebbR. A.ButnerH. M.GearW. K.WalkerH. J.1998A dust ring around Eridani: analog to the young Solar SystemAstrophys. J.50613313710.1086/311652Open DOISearch in Google Scholar

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:10.1086/308942JayawardhanaR.HollandW. S.GreavesJ. S.DentW. R. F.MarcyG. W.HartmannL. W.FazioG. G.2000Dust in the 55 Cancri planetary systemThe Astrophysical Journal53642542810.1086/308942Open DOISearch in Google Scholar

Jiang I. G., Yeh L. C (2003), Bifurcation for dynamical systems of planet-belt interaction, Int. J. Bifurc. Chaos 13(3): 617 – 630. Doi:10.1142/s0218127403006807.JiangI. G.YehL. C2003Bifurcation for dynamical systems of planet-belt interactionInt. J. Bifurc. Chaos13361763010.1142/s0218127403006807Open DOISearch in Google Scholar

Jiang, I.G., Yeh, L.C. (2004b), On the chaotic orbits of disk-star-planet systems, Astron. J. 128: 923 – 932. Doi:10.1086/422018.JiangI.G.YehL.C.2004bOn the chaotic orbits of disk-star-planet systemsAstron. J.12892393210.1086/422018Open DOISearch in Google Scholar

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:10.1111/j.1365-2966.2004.08504.x.JiangI. G.YehL. C.2004aThe drag-induced resonant capture for Kuiper belt objectsMon. Not. R. Astron. Soc.355293210.1111/j.1365-2966.2004.08504.xOpen DOISearch in Google Scholar

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.JiangY.BaoyinH.2019Periodic orbits related to the equilibrium points in the potential of Irregular–shaped minor celestial bodiesResults in Physics12368374Doi.org/10.1016/j.rinp.2018.11.049.Search in Google Scholar

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.2015.06.005JungC.ZotosE. E.2015Order and chaos in a three dimensional galaxy modelMechanics Research Communications694553Doi.org/10.1016/j.mechrescom.2015.06.005Search in Google Scholar

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:10.1007/s10509-008-9823-6KushvahB.S.2008The effect of radiation pressure on the equilibrium points in the generalised photogravitational restricted three body problemAstrophys. Space Sci.31523124110.1007/s10509-008-9823-6Open DOISearch in Google Scholar

Miyamoto M., Nagai R. (1975), Three-dimensional models for the distribution of mass in galaxies, Publ. Astron. Soc. Jpn. 27: 533 – 543.MiyamotoM.NagaiR.1975Three-dimensional models for the distribution of mass in galaxiesPubl. Astron. Soc. Jpn.27533543Search in Google Scholar

Papadakis K.E. (2008), Families of asymmetric periodic orbits in the restricted three-body problem, Earth, Moon and Planets, 103: 25 – 42. Doi: 10.1007/s11038-008-9232-4PapadakisK.E.2008Families of asymmetric periodic orbits in the restricted three-body problemEarth, Moon and Planets103254210.1007/s11038-008-9232-4Open DOISearch in Google Scholar

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: 10.1007/s40295-019-00178-zPathakN.AbouelmagdE. I.ThomasV. O.2019On higher order of resonant periodic orbits in the photogravitational restricted three body problemJ of Astronaut Sci66447550510.1007/s40295-019-00178-zOpen DOISearch in Google Scholar

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: 10.3934/dcdss.2019057PathakN.ThomasV. O.AbouelmagdE. I.2019The photogravitational restricted three-body problem: Analysis of resonant periodic orbitsDiscrete and Continuous Dynamical Systems - Series S (DCDS-S)124&584987510.3934/dcdss.2019057Open DOISearch in Google Scholar

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:10.1017/S1743921315008388PitjevaE. V.PitjevN. P.2016Masses of asteroids and total mass of the main asteroid beltInternational Astronomical Union31821221710.1017/S1743921315008388Open DOISearch in Google Scholar

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: 10.3934/dcdss.2019044SelimH. H.GuiraoJ. L. G.AbouelmagdE. I.2019Libration points in the restricted three–body problem: Euler angles, existence and stabilityDiscrete and Continuous Dynamical Systems - Series S (DCDS-S)124&570371010.3934/dcdss.2019044Open DOISearch in Google Scholar

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:10.1007/s10569-011-9338-2ShibayamaM.YagasakiK.2011Families of symmetric relative periodic orbits Originating from the circular Euler solution in the isosceles three–body ProblemCelest. Mech. Dyn. Astron.110537010.1007/s10569-011-9338-2Open DOISearch in Google Scholar

Singh J., Begha J. M.(2011), Periodic orbits in the generalized perturbed restricted three-body problem, Astrophys. Space Sci. 332: 319 – 324. Doi: 10.1007/s10509-010-0545-1SinghJ.BeghaJ. M.2011Periodic orbits in the generalized perturbed restricted three-body problemAstrophys. Space Sci.33231932410.1007/s10509-010-0545-1Open DOISearch in Google Scholar

Singh J., Taura J.J. (2013), Motion in the generalized restricted three-body problem, Astrophys. Space Sci. 343(1): 95 – 106. Doi:10.1007/s10509-012-1225-0.SinghJ.TauraJ.J.2013Motion in the generalized restricted three-body problemAstrophys. Space Sci.34319510610.1007/s10509-012-1225-0Open DOISearch in Google Scholar

Szebehely V. (1967), Theory of Orbits: The Restricted Problem of Three BodiesAcademic Press, New YorkSzebehelyV.1967Theory of Orbits: The Restricted Problem of Three BodiesAcademic PressNew YorkSearch in Google Scholar

Yeh L. C., Jiang I. G. (2006), On the Chermnykh-like problems: II. The equilibrium points, Astrophys. Space Sci. 306: 189 – 200. Doi:10.1007/s10509-006-9170-4.YehL. C.JiangI. G.2006On the Chermnykh-like problems: II. The equilibrium pointsAstrophys. Space Sci.30618920010.1007/s10509-006-9170-4Open DOISearch in Google Scholar

Plan your remote conference with Sciendo