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.
- Restricted three bodies problem
- Asteroids belt effect
- Linear Stability
- Periodic orbits
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 .
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,  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.  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
 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.  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
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 . 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 . They demonstrated that there are two new equilibrium points (
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.
 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 . 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  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  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.
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:
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 . This model is called flattened potential and used in modelling disk galaxies. It can be controlled by
The parameter The parameter
Hence the perturbed acceleration regard to the asteroid belt is
We assume that
Now we normalize the units as the sum of two masses
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
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
In the case of collinear equilibrium points (
The equilibrium point
Since the point
Now using the Lagrangian inversion method to inverting the above series, we get
In the case of the triangular equilibrium points (
Now we will keep the linear terms
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 (
At the collinear libration points
At the triangular points, we have
Since 0 <
Eqs. (44, 45) show that the discriminant If 0 < If When
If 0 <
Under the previous discussion, when Eq. (43) is equal zero, then one can obtain the value of critical mass (
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
From Eqs. (52) we get the velocity variation in the form
The triangular points are linearly stable in the range 0 <
Either the long or short period terms can be eliminated from the solving by properly selected initial conditions. The four initial conditions at
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 (
Since Eq. (60) includes bilinear term
Furthermore the lengths of semi–major (
While the periodic of motion
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.