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

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.


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 J 4 . 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: htt ps : //en.wikipedia.org/wiki/Asteroid b elt). 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 where • M b is the total mass of the disc.
• r is the radial distance of the infinitesimal body it is given by r 2 = x 2 + y 2 .
• 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 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 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

Model description
We assume that m 1 and m 2 denote the bigger and smaller primaries masses respectively, and m is the mass of the infinitesimal body. We consider both masses m 1 and m 2 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 OXY Z 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 r 1 and r 2 be the distances between m and the primaries m 1 and m 2 respectively, while R the separation distance between m 1 and m 2 . The coordinates of m 1 , m 2 and m are (x 1 , 0, 0) , (x 2 , 0, 0) and (x, y, 0) respectively. Now we normalize the units as the sum of two masses m 1 and m 2 is one and the distance between them also is taken as one. In addition the gravitational constant is one. We also assume that µ = m 2 /(m 1 + m 2 ) be the mass parameter. Consequently m 2 = µ and m 1 = 1 − µ with m 1 > m 2 and 0 < µ ≤ 1/2. Then in the XY −plane, the coordinates of m 1 ,m 2 and m are (x 1 , 0, 0) = (µ, 0, 0), (x 2 , 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 m 1 and m 2 are given bÿ In Eq. (5), the angular velocity n = 1, see for details [35], where 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 bÿ where Ω = and r 1 and r 2 are given by Eq. (6), while the perturbed mean motion n is here in Eq. (9), r 2 c = 1 − µ + µ 2 , see [34] for details. Using Eqs. (7,8) then the Jacobian integral is governed bẏ 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 whenẍ =ÿ =ẋ =ẏ = 0, therefore Ω x = Ω y = 0, then we have to solve the following two equations 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 (L 1 , L 2 and L 3 ) y = 0, so that the equilibrium points lie on the line joining the primaries (X−axis), see Fig. 2, so we have The equilibrium point L 1 lies beyond masse m 1 as in Fig. 2. Then in this case, we have Let r 2 = ξ 1 , r 1 = 1 + ξ 1 , x 1 = µ − 1 − ξ 1 , and n 2 = s, then with a help of Eq. (12), we get after writing Eq. (13) in the series form, we get Then the parameter µ as a function in the variation ξ 1 is given by Consequently, we will using the Lagrangian inversion method to inverting the above series, which represents µ, to express ξ 1 as functions of µ, then we get hence

Location of L 3
The point L 3 lies beyond the large mass according to Fig. 2, r 2 − r 1 = 1, r 1 = x − µr 2 = x − µ + 1 and ∂ r 1 /∂ x = ∂ r 2 /∂ x = 1, then Eq. (11) can be rewritten in the form after substituting r 2 = ξ 3 + 2 and r 1 = ξ 3 + 1, into Eq. (20), we get Again the form series of Eq. (21) is given by Then µ as a function series in ξ 3 is given by Now using the Lagrangian inversion method to inverting the above series, we get 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 (L 4 and L 5 ) y = 0 and Ω x = 0 = Ω y . Using Eqs. (10), we get 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 r 1 = r 2 = 1 [35]. But, due to the perturbations from the belt, we assume that the solutions may change slightly by ε, then we can write From Eqs. (9, 23, 24), we get where ε is very small quantities represents perturbation effect of asteroids belt. Now we will keep the linear terms ε and neglecting the higher orders, then with a help of Eq. (25), we have From Eqs. (6,24,26), we get the locations of the triangular points in the following form

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 (x 0 , y 0 ), we employ small displacement (ξ , η) to the coordinate (x 0 , y 0 ) where (x 0 , y 0 ) represents the coordinates of one of five equilibria points. So that the vector of variation is related to the initial stat vector by r = r 0 + ∆r where r 0 ≡ (x 0 , y 0 ), r ≡ (x, y) and ∆r ≡ (ξ , η), then we can write We linearize the equations of motion by using Taylor series around equilibrium. Then using Eqs. (7) and Eqs. (28), we obtainξ From the above equations the linear variational equations arë 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 (x 0 , y 0 ), where these derivatives are given by Now we suppose that the solutions of the Eqs. (29) are where K, M and λ constant, therebẏ ξ = kλ e λt ,ξ = kλ 2 e λt ,η = Mλ e λt ,η = Mλ 2 e λt .
substituting Eqs. (31,32) into Eqs. (29), then the characteristic equation is determined by hence, we get The equilibrium point (x 0 , y 0 ) 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 r 1 = |x − µ| and r 2 = |x − µ + 1|. From Eqs. (30,33), the characteristic equation in case of collinear libration points can be reduced to Let Λ = λ 2 , Eq. (34) can be rewritten in the form then Eq. (35) represents a quadratic equation, which its solution can be written in the form 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 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 and substituting λ 2 = ω into the characteristic Eq. (33), then we get where a and c can be evaluated with a help of Eqs. (38) and the following two relations Thereby the roots of Eq. (39) are given by which can be written as of a function of the mass parameter µ in the form here α and β are the coefficients µ 2 and µ respectively. Using Eq. (41) the roots are given by Since 0 < µ ≤ 1 2, then with using Eq. (43), we can study the behavior of D in the interval 0, 1 2 2(r 2 c + T 2 ) 3/2 + 3M b 8r 2 c + 9 4(r 2 c + T 2 ) 5/2 < 0 , when µ = 1/2 Also the derivative of D with respect to the parameter µ is given by because M b 1, and using Eqs. (44, 45), we obtain 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.
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.

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 where utilizing Eqs. (46, 47) we get Eq. (48) represents the value of the critical mass with two parts. The first µ 0 ) 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 = µ 0 , where the values of µ 0 and µ p are given by the following relations 6 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 K j and M j , it is governed by This relation means that the coefficients K j and M j ( j = 1, 2, 3, 4) are dependent. Therefore the four initial conditions ξ 0 , η 0 ,ξ 0 andη 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 = t 0 ). Substituting Eq. (49) into Eq. (31), we get The coefficient can be expressed as function of the initial conditions, because the determinant (∆) of System of Eqs. (50) is not zero.
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 Now We can take K 1 = K 2 = 0, therefor the solution in Eqs. (51) can rewritten in the following form where the real quantities τ and β 3 are defined by From Eqs. (52)we get the velocity variation in the forṁ Using Eqs. (53), when t = t 0 , one obtainṡ With a help of Eq. (54) we can eliminate the time from Eq. (52), and the equation of periodic orbits reduced to which can be rewritten in the following standard form 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 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. Sinceη 0 = −ξ 0 β 3 τ,ξ 0 = 0 at ξ 0 = 0 and η 0 = 0, the motion along the orbits is retrograde.

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 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 =C 1 cos s 1 t +D 1 sin s 1 t +C 2 cos s 2 t +D 2 sin s 2 t.
(56) Therefor the terms with coefficients C 1 , D 1 ,C 1 ,D 1 associated with angular frequencies s 1 (mean motion) refer to the long periodic orbits and terms with coefficients C 2 , D 2 ,C 2 ,D 2 associated with angular frequencies s 2 , which refer to the short periodic orbits. In addition s 2 1,2 = −ω, thereby we get Substituting Eqs. (42 , 40) into Eq. (57), with some simple calculations, the angular frequencies of motion s 1 and s 2 are given by 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 1, 2) and Ω 0 xx , Ω 0 xy and Ω 0 yy 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 ( ξ 0 , η 0 ,ξ 0 andη 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 =C 2 =D 2 = 0, then the relation between the initial conditions and the coefficients in Eqs. (56)are

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 In which the negative sign (plus) means that the infinitesimal body starts its motion from L 4 (L 5 ). 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 =C 1 cos s 1 t + D 1 sin s 1 t, η =C 1 cos s 1 t +D 1 sin s 1 t. 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 called normal coordinates such that the bilinear term dose not appear. Hence the old and new coordinates system are related by the following equation where plus sign (minus sign) refers to the center of ellipse at L 4 (L 5 ).

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.