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 J4. In addition  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 . 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 (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.
 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.
2 Background on asteroids belt potential
In the solar system, the asteroid belt is similar to a ring-shaped. it can found between the Mars and Jupiter orbits. This region includes many objects (minor planets) with different sizes and shapes, which are irregular in most cases but very smaller than compered to the planets. In particularly, this belt is called the main asteroids belt, in order to characterize it from any other collection of asteroids in the solar system, such as trojan or near–earth asteroids, see Fig.1 (Source: https://en.wikipedia.org/wiki/Asteroidbelt). The asteroid belt region lies between the range of radial distances from 2.06 to 3.27 AU. It includes about 93.4% minor planets. These distances represent the inner and outer boundaries of the main belt region respectively . 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 .
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.
- 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.
3 Model description
We assume that m1 and m2 denote the bigger and smaller primaries masses respectively, and m is the mass of the infinitesimal body. We consider both masses m1 and m2 move in circular orbits around their common center of mass. Furthermore the infinitesimal body m moves in the same plane of primaries motion under their mutual gravitational fields. We also assume that the coordinate system OXYZ rotates about OZ–axes by the angular velocity n in positive direction. OX–axis is taken the joining line between the primaries, OY – axis is perpendicular to OX–axis and OZ–axis is perpendicular to the orbital plane of the primaries. Let r1 and r2 be the distances between m and the primaries m1 and m2 respectively, while R the separation distance between m1 and m2. The coordinates of m1, m2 and m are (x1,0,0), (x2,0,0) and (x,y,0) respectively.
4 Locations of equilibrium points
4.1 Location of collinear points
4.1.1 Location of L1
4.1.2 Location of L2
4.1.3 Location of L3
4.2 Location of triangular points
5 Stability of motion around the libration point
5.1 Stability of collinear points
5.2 Stability of triangular points
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,
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.
5.3 Critical mass
6 Periodic orbits
6.1 Periodic orbits around collinear points
6.2 Periodic orbits around triangular points
6.2.1 Elliptic orbits
6.2.2 The orientation of principal axes of the ellipse
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).
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.
The Fifth authors (EIA) is partially supported by Fundación Séneca (Spain), grant 20783/PI/18, and Ministry of Science, Innovation and Universities (Spain), grant PGC2018 - 097198 - B -100
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:
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:
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
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:
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 :
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
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:
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:
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
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:
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:
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:
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:
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:
Elipe A., Lara M.(1997), Periodic orbits in the restricted three body problem with radiation pressure, Celest. Mech. Dyn. Astron. 68: 1 – 11.
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:
- Export Citation
Elshaboury S. M., Abouelmagd E. I., Kalantonis V. S., Perdios E. A. (2016),)| false 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-x
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:
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:
- Export Citation
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),)| false A dust ring around Eridani: analog to the young Solar System, Astrophys. J. 506: 133 – 137. Doi: 10.1086/311652
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:
Jiang I. G., Yeh L. C (2003), Bifurcation for dynamical systems of planet-belt interaction, Int. J. Bifurc. Chaos 13(3): 617 – 630. Doi:.
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:.
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.
Jung C., Zotos E. E. (2015), Order and chaos in a three dimensional galaxy model, Mechanics Research Communications 69: 45 – 53 Doi.org/10.1016/j.mechrescom.2015.06.005
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:
Miyamoto M., Nagai R. (1975), Three-dimensional models for the distribution of mass in galaxies, Publ. Astron. Soc. Jpn. 27: 533 – 543.
Papadakis K.E. (2008), Families of asymmetric periodic orbits in the restricted three-body problem, Earth, Moon and Planets, 103: 25 – 42. Doi:
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:
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:
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:
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:
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:
Singh J., Begha J. M.(2011), Periodic orbits in the generalized perturbed restricted three-body problem, Astrophys. Space Sci. 332: 319 – 324. Doi:
Singh J., Taura J.J. (2013), Motion in the generalized restricted three-body problem, Astrophys. Space Sci. 343(1): 95 – 106. Doi:.
Szebehely V. (1967), Theory of Orbits: The Restricted Problem of Three BodiesAcademic Press, New York
Yeh L. C., Jiang I. G. (2006), On the Chermnykh-like problems: II. The equilibrium points, Astrophys. Space Sci. 306: 189 – 200. Doi:.