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.

#### Keywords

- Jupiter-Europa system
- Oblateness
- Equilibrium point
- Linear stability test
- Tadpole and horseshoe orbits
- Poincaré surface of section

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 [25]. 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 [1] 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. [8] 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 _{4} and _{5} in the Sun-Jupiter system can be more understandable by studying the tadpole and horseshoe orbits about the _{4} and _{5} points, respectively [9, 16]. Rabe [22] and Taylor [26] have determined approximate initial conditions to compute a series of periodic orbits around the _{4} and _{5} points of the Sun-Jupiter system and they have found stable horseshoe-shaped periodic orbits.

In order to understand and to analyse the evolution of periodic orbits, Winter [27] 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. [5] 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 [3] 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.

Let _{1} and _{2} be the masses of bigger and smaller oblate primaries, respectively. Assume that the primaries rotate about their common centre of mass in the _{1} + _{2}) = 1. Since the primaries are considered as oblate spheroid therefore, the angular speed (mean motion) _{2i}, _{1} and _{2} of the infinitesimal mass from the bigger and smaller primaries are defined as

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_{x} = 0, Ω_{y} = 0, simultaneously i.e.

_{1},

_{2},

_{3}

Collinear equilibrium points are those points, which lie on the line joining the primaries. Therefore, for collinear equilibrium points _{1} = |_{2} = |

In the case of collinear equilibrium points, Eq. (6) is unimportant due to

For simplicity, collinear axis is divided into three parts as _{1}, _{2} and _{3}, respectively.

In case of _{1} point (i.e for region −

Similarly, in case of _{2}: _{3}: _{1}, _{2} and _{3}, respectively. We have computed the collinear equilibrium points _{1}, _{2} and _{3} by varying the oblateness coefficient _{21} and _{22} in Jupiter-Europa system as shown in Table 1. From Table 1 it is observed that on increment in the oblateness coefficient _{21} of the Jupiter shows more variation in the position of _{1}, but a small effect is seen both in case of _{2} and _{3} points. Due to the oblateness coefficient _{22} of Europa, we found a considerable change in the position of _{1}, _{2} and _{3} points. An increment in the value of _{22} corresponds to a shift in the position of _{1} and _{2} points towards Europa, but _{3} slightly move away from the Jupiter. Further, points _{1} and _{3} move towards Jupiter on increment in the oblateness _{22} of Europa while _{2} moves away from the Europa.

Position of _{1}, _{2}, _{3}, _{4} and _{5} in Jupiter-Europa system at

_{21} |
_{22} |
_{1} |
_{2} |
_{3} |
_{4,5} : ( |
---|---|---|---|---|---|

0 | 0 | 0.9798121 | 1.0204124 | −1.0000104 | 0.4999749, 0.8660254 |

0.0001 | 0 | 0.9798138 | 1.0204108 | −1.0000104 | 0.5000249, 0.8659965 |

0.0002 | 0 | 0.9798155 | 1.0204091 | −1.0000105 | 0.5000749, 0.8659677 |

0.0003 | 0 | 0.9798172 | 1.0204074 | −1.0000105 | 0.5001248, 0.8659388 |

0 | 0.0035 | 0.9646568 | 1.0350740 | −0.9982666 | 0.4982325, 0.8650171 |

0 | 0.0070 | 0.9597175 | 1.0394326 | −0.9965348 | 0.4965052, 0.8640129 |

0 | 0.0105 | 0.9563164 | 1.0422269 | −0.9948149 | 0.4947928, 0.8630128 |

0.000285 | 0.007198 | 0.9595035^{*} |
1.0396141^{*} |
−0.9964397^{*} |
0.4965504^{*}, 0.8638773^{*} |

represent the values are at actual oblateness parameters

_{4},

_{5}

Since, for triangular equilibrium points _{1} = _{2} = 1 which implies
_{1} = 1 + _{1} and _{2} = 1 + _{2}, where _{1}, _{2} are very small real numbers. Substituting the value of _{1} and _{2} in Eqs (5) and (6) then, solving for _{1} and _{2} and their product, we get

The expression for _{1} and _{2} are obtained by solving the following system of equations

The system of Eqs (13) and (14) are achieved by putting the value of _{1} and _{2}, where

Thus, substituting the value of _{1} and _{2} in Eqs (11) and (12) we get triangular equilibrium points _{4}; _{5}. In Table 1, apart from collinear equilibrium points, we have also presented the positions of triangular equilibrium points _{4} and _{5} at different values of oblateness coefficient _{21} and _{22} of the primaries. From Table 1, it is observed that the effect of the oblateness coefficient _{21} of the Jupiter causes large variation in the positions of _{4,5}. However the oblateness coefficient _{22} of Europa influence more in the positions of _{4} and _{5} points compared to that of _{22}. A combined effect of the approximate value of oblateness coefficients _{21}, _{22} on equilibrium points _{1},_{2}, _{3}, _{4} and _{5} have also been obtained as shown in bottom row of Table 1.

Let the co-ordinate of the equilibrium point be denoted by (_{0},_{0}). Suppose _{0},_{0}), then position of infinitesimal mass at any future time is given as _{0} + _{0} + _{x} and Ω_{y} about the equilibrium point (_{0}, _{0}) up to linear order terms in

Therefore, the equations of motion (1) and (2) in the neighbourhood of (_{0},_{0}) can be written by using Taylor series expansion in the linear order as
_{0}, _{0}) and the expressions of Ω_{xx}, Ω_{xy}, Ω_{yy} are given by

Now, we investigate the motion of infinitesimal body about the equilibrium point (_{0}, _{0}) in ^{λ t}^{λ t}

The above equations have a non-trivial solution for _{21} and _{22}. For the linear stability of equilibrium points, the nature of roots play a key role, which depends on the sign of discriminant
_{j}_{j}

For the collinear equilibrium points

Thus, solutions of Eqs (15) and (16) can be written as

The exponential factors ^{at}^{−at} in Eqs (21) and (22) results in unbounded motion in the _{1}, _{2} and _{3} remain unstable for all 0

In case of triangular equilibrium points,

In absence of the oblateness, i.e. when _{21} = 0, _{22} = 0, the Eq. (23) gives the critical mass ratio _{c}_{21} = 0.000285, _{22} = 0.007189), the critical value of mass ratio becomes _{c}

^{2} − 4

Let

_{c}^{2} − 4

This case implies

_{c}^{2} − 4

In this case, we found that the roots are of the form

Further, we found that all characteristic roots at _{21} = 0.000285, _{22} = 0.007198 are purely imaginary for mass ratio 0 _{4,5} are stable there. Also, we have observed a variation in the nature of characteristic roots with respect to oblateness _{21}, _{22} as given in Figure 3 and noticed that the oblateness of Europa affects more than that of oblateness of Jupiter on the stability region.

Suppose, the coefficients _{j}_{j}_{j}_{j}_{j}_{j}_{j}_{j}

Therefore, after simplification solutions (19) and (20) reduces to

In particular at _{4} (0.4965504, 0.8638773), _{21} = 0.000285 and _{22} = 0.007198, the perturbed solution ^{−4}, ^{−4} and
_{1,2} _{3,4} _{4,5} are stable. The semi analytical solution

In the RTBP, there exist different types of motion of an infinitesimal mass in the vicinity of the _{4,5}, valid only for small magnitude of displacement from the equilibrium point. Rabe [22] has computed the approximate initial conditions to analyse the periodic orbits close to _{4,5}. He has found two set of initial conditions for horseshoe orbits and for long tadpole-shaped orbits around the _{4,5}. Also, Taylor [26] briefly has described the smooth horseshoe orbits around _{4,5} for different initial conditions in the Sun-Jupiter system without perturbation. We are interested to compute the tadpole and horseshoe periodic orbits in the Jupiter-Europa system to analyse the effect of oblateness on these orbits. To determine the tadpole and horseshoe orbits in Jupiter-Europa system, we have followed the method described in [16, 26].

We assume that the infinitesimal mass _{4} or _{5}, which is the closest approach for _{4} or _{5}. A good approximation for periodic orbit about _{4,5} is that the magnitude of velocity _{4} be _{4} and _{5} in the Jupiter-Europa system in presence and absence of oblateness of Jupiter and Europa. For the initial conditions corresponding to the value of _{4} (also, similar tadpole orbits about _{5} are obtained which are not presented here) at _{21} = _{22} = 0. We have analysed the effect of oblateness on tadpole and horseshoe orbits as plotted in Figures 5(b,c,d) and 6(b,c,d). We have found that tadpole orbits gets elongated gradually due to oblateness of Jupiter (_{21} = 0.0285), but a slight increment in the oblateness of Europa (_{22} = 0.0007189) results in more elongation of the tadpole orbits (Figures 5b, 5c, 5d) with small deviation in amplitude of loops. On the other hand, in Figure 6(a–d), the horseshoe orbits about _{4} and _{5} at

The position coordinate (

We have computed the PSS at different values of mass ratio _{j}_{21}, _{22} by the method described in [7,16,27]. The starting value of _{j}_{j}_{j}_{j}_{j}_{21} and _{22} varies, the structural region and islands changes in character and one particular structure may become more dominant as shown in Figures 9 and 10, but reduction in the chaotic zones is more due to oblateness of smaller primary _{22} than that of bigger primary _{21}. In presence of oblateness, it is found that the curves or chain of islands representing quasi-periodic orbits predominates the chaotic regions.

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 < _{c}_{c}_{1,2,3,4,5} have shown deviations from their respective positions in classical cases (Table 1). Again, we have reckoned tadpole and horseshoe orbits in the Jupiter-Europa system under the influence of oblateness for different values of _{4} at = 0.0000251 and _{21} = _{22} = 0. The effect of oblateness on tadpole and horseshoe orbits are analyzed (Figures 5b, 5c, 5d and 6b, 6c, 6d) and it is observed that tadpole orbits gets elongate gradually due to oblateness of Jupiter (_{21} = 0.0285), but a slight increment in the oblateness of Europa (_{22} = 0.0007189) results in more elongation of the tadpole orbits (Figures 5b, 5c, 5d) with small deviation in amplitude of loops. On the other hand, from the horseshoe orbits about _{4} and _{5} at _{j}_{21} and _{22} (Figures 7–10). It is noticed that the oblateness of the primaries minimize the chaotic regions and induce the formation of a chain of islands that corresponds to quasi-periodic orbits. Also, we have found that the oblateness parameters _{21} and _{22} affect the structure and nature of orbits. The PSS in our model can be degraded to that of PSS without the oblateness of primaries [7]. Finally, it is concluded that shape and size of the primaries play a significant role in the study of motion of the infinitesimal mass. These results may help to the study of more generalized problem of three bodies in space.

#### Position of L1, L2, L3, L4 and L5 in Jupiter-Europa system at μ = 0.0000251

_{21} |
_{22} |
_{1} |
_{2} |
_{3} |
_{4,5} : ( |
---|---|---|---|---|---|

0 | 0 | 0.9798121 | 1.0204124 | −1.0000104 | 0.4999749, 0.8660254 |

0.0001 | 0 | 0.9798138 | 1.0204108 | −1.0000104 | 0.5000249, 0.8659965 |

0.0002 | 0 | 0.9798155 | 1.0204091 | −1.0000105 | 0.5000749, 0.8659677 |

0.0003 | 0 | 0.9798172 | 1.0204074 | −1.0000105 | 0.5001248, 0.8659388 |

0 | 0.0035 | 0.9646568 | 1.0350740 | −0.9982666 | 0.4982325, 0.8650171 |

0 | 0.0070 | 0.9597175 | 1.0394326 | −0.9965348 | 0.4965052, 0.8640129 |

0 | 0.0105 | 0.9563164 | 1.0422269 | −0.9948149 | 0.4947928, 0.8630128 |

0.000285 | 0.007198 | 0.9595035^{*} |
1.0396141^{*} |
−0.9964397^{*} |
0.4965504^{*}^{*} |

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