The study of motion of a test mass in the vicinity of an equilibrium point under the frame of restricted three body problem (RTBP) plays an important role in the trajectory design for different space missions. In this paper, motion of an infinitesimal mass has been described under the frame of Jupiter-Europa system with oblateness. At first, we have determined equilibrium points and then performed linear stability tests under the influence of oblateness of both the primaries. We found that due to oblateness, a considerable deviation in the existing results has occurred. Next, we have computed tadpole and horseshoe orbits in the neighbourhood of triangular equilibrium points and then the oblateness effect is recorded on these orbits. Finally, the evolution of orbits of infinitesimal mass about triangular equilibrium points have been estimated by using Poincaré surface of section technique and it is noticed that in presence of oblateness, quasi-periodic orbit dominates over the chaotic zones. These results will help in further study of more generalised models with perturbations.
The restricted three body problem (RTBP) describes the motion of a restricted mass (also known as infinitesimal mass), which is moving under the effect of gravitational field of two massive bodies (called as primaries), without disturbing the actual motion of these primaries about their common centre of mass . The RTBP is a demonstrative model of many problems in celestial mechanics, and it is used as a base model to describe the dynamics of small bodies (asteroid, comet, satellite, spacecraft, etc.) in solar as well as in extra-solar planetary systems [3, 11].
In the study of RTBP, the perturbing forces such as radiation pressure, Poynting-Robertson drag, solar wind drag, oblateness of massive bodies, etc. have an effective influence on celestial bodies. A number of researchers have contributed to the study of motion of infinitesimal mass under the frame of RTBP, Chermnykh-like problem etc. through different approaches and techniques in the presence of different perturbations in the form of oblateness, radiation pressure force, disc's effect, drag forces etc. [4, 6, 12, 13, 24, 29]. AbdulRaheem and Singh  have studied the combined effect of Coriolis force, centrifugal force, radiation pressure force and oblateness of the primaries on the equilibrium points and found that only Coriolis force acts as a stabilising factor, whereas centrifugal force, radiation pressure force and oblateness act as destabilising factors that cause a contraction in the stability range of the triangular equilibrium points. A few authors [15, 23, 28] have discussed the effects of radiation pressure, triaxial rigid bodies, oblateness of massive bodies and presence of a disc during the analysis of equilibrium points and their stability. They have found a deviation in the positions of equilibrium points and stability region because of these perturbations.
In the solar system dynamics, the analysis of periodic orbits produces many important ideas to understand the general properties of different dynamical systems. Companys et al.  have determined the existence of quasi-periodic orbits near the equilibrium points of the Earth-Moon system. The quasi-periodic orbits play an essential role in the designing of new space missions due to which fuel consumption in the spacecraft reduces. Motion of asteroids in the surroundings of triangular equilibrium points
In order to understand and to analyse the evolution of periodic orbits, Winter  studied the evolution of periodic and quasi-periodic orbits and their stability property in the phase space of the Earth-Moon system using a numerical technique known as Poincaré surfaces of section (PSS) and found that these are sensitive dependence on the initial conditions. Many authors [10, 14, 21] have analysed the nature and stability of the periodic and quasi-periodic orbits with the help of PSS under the frame of perturbed restricted three-body problem and have noticed the effects of perturbing parameters. Abouelmagd et al.  have investigated the analytical derivations to study periodic solutions for the two-body problem perturbed by the first zonal harmonic parameter by using different methods. Further, Abouelmagd and Ansari  have analysed the cases, which agree that Jacobian and energy conversation are constants, in the Sun-perturbed Earth-Moon system and have illustrated the equilibrium points and their stability and PSS. Again, Pathak et al. [17, 19] have studied interior higher order resonant orbits in the perturbed photo-gravitational restricted three-body problem and have found that there exist periodic orbits for seventh- and ninth-order resonance, which are passing around the Earth. They also analysed the effect of radiation pressure and oblateness on these orbits.
In this paper, we are interested to determine the tadpole and horseshoe orbits and to observe the dynamical behaviour of orbits using PSS technique under the frame of RTBP with oblate primaries. The paper is organised as follows: Section 2 presents the equations of motion of the infinitesimal mass in the presence of oblateness. Sections 3 and 4 are devoted to the determination of equilibrium points and their linear stability test under the effect of oblateness, respectively. Tadpole and horseshoe orbits in the neighbourhood of equilibrium points are discussed in Section 5. Computation of PSS and the analysis of the effect of perturbing parameters on the PSS are presented in Section 6. Finally, the paper is concluded in Section 7. Numerical and semi-analytic computations are performed by using MATHEMATICA software (Version 12.0). Accuracy and precision goals of the values are achieved by considering or taking the values up to seven decimal places.
Now, multiplying Eq. (1) by
On integration, we obtain
In this section, we shall discuss the existence of those points at which the velocity of the infinitesimal mass become zero. These points can be obtained by solving Ω
Collinear equilibrium points are those points, which lie on the line joining the primaries. Therefore, for collinear equilibrium points
In the case of collinear equilibrium points, Eq. (6) is unimportant due to
For simplicity, collinear axis is divided into three parts as
In case of
Similarly, in case of
represent the values are at actual oblateness parameters
Since, for triangular equilibrium points
The expression for
The system of Eqs (13) and (14) are achieved by putting the value of
Thus, substituting the value of
Let the co-ordinate of the equilibrium point be denoted by (
Therefore, the equations of motion (1) and (2) in the neighbourhood of (
Now, we investigate the motion of infinitesimal body about the equilibrium point (
The above equations have a non-trivial solution for
For the collinear equilibrium points
Thus, solutions of Eqs (15) and (16) can be written as
In case of triangular equilibrium points,
In absence of the oblateness, i.e. when
This case implies
In this case, we found that the roots are of the form
Further, we found that all characteristic roots at
Suppose, the coefficients
Therefore, after simplification solutions (19) and (20) reduces to
In particular at
In the RTBP, there exist different types of motion of an infinitesimal mass in the vicinity of the
We assume that the infinitesimal mass
The position coordinate (
We have computed the PSS at different values of mass ratio
We have considered the Jupiter-Europa system with oblateness of Jupiter and Europa and have found the equations of motion of infinitesimal mass. We have determined the equilibrium points and examined its linear stability under the influence of oblateness of both the primaries. It is found that due to oblateness of Jupiter and Europa, the stability range of the triangular equilibrium point is deviated from 0 <
Position of L1, L2, L3, L4 and L5 in Jupiter-Europa system at μ = 0.0000251