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Motion about equilibrium points in the Jupiter-Europa system with oblateness

Data publikacji: 15 Apr 2022
Tom & Zeszyt: AHEAD OF PRINT
Zakres stron: -
Otrzymano: 14 Sep 2020
Przyjęty: 01 Jun 2021
Informacje o czasopiśmie
License
Format
Czasopismo
eISSN
2444-8656
Pierwsze wydanie
01 Jan 2016
Częstotliwość wydawania
2 razy w roku
Języki
Angielski
Abstract

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

Introduction

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 L4 and L5 in the Sun-Jupiter system can be more understandable by studying the tadpole and horseshoe orbits about the L4 and L5 points, respectively [9, 16]. Rabe [22] and Taylor [26] have determined approximate initial conditions to compute a series of periodic orbits around the L4 and L5 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.

Equations of Motion

Let m1 and m2 be the masses of bigger and smaller oblate primaries, respectively. Assume that the primaries rotate about their common centre of mass in the xy-plane due to mutual gravitational attraction and the infinitesimal mass moves under the gravitational field of the primaries in the orbital plane without influencing their motion as shown in Figure 1. Assume that the sum of masses of the primaries is the unit of mass and distance between the primaries is the unit of distance. The unit of time is such that G(m1 + m2) = 1. Since the primaries are considered as oblate spheroid therefore, the angular speed (mean motion) n is written [2, 28] as n=1+3(A21+A22)2 n = \sqrt {1 + {{3({A_{21}} + {A_{22}})} \over 2}} , where A2i, i = 1, 2 represent the oblateness coefficients of the bigger and smaller primary [28]. The mass ratio is defined as μ=m2m1+m2 \mu = {{{m_2}} \over {{m_1} + {m_2}}} so that 0<μ12 0 < \mu \le {1 \over 2} . Let the coordinates of the bigger and smaller primary be (−μ, 0) and (1 − μ, 0), respectively and (x, y) be the position coordinate of the infinitesimal mass in the synodic frame Oxy. Therefore, the equations of motion of the infinitesimal mass P(x, y) in the xy-plane [18, 25] are written as x¨2ny˙=Ωx, \ddot x - 2n\dot y = {\Omega _x}, y¨+2nx˙=Ωy, \ddot y + 2n\dot x = {\Omega _y}, where the effective potential Ω, experienced by the infinitesimal mass is given as Ω=n22(x2+y2)+(1μ)r1(1+A212r12)+μr2(1+A222r22) \Omega = {{{n^2}} \over 2}({x^2} + {y^2}) + {{(1 - \mu)} \over {{r_1}}}\left({1 + {{{A_{21}}} \over {2{\kern 1pt} r_1^2}}} \right) + {\mu \over {{r_2}}}\left({1 + {{{A_{22}}} \over {2{\kern 1pt} r_2^2}}} \right) and the respective distances r1 and r2 of the infinitesimal mass from the bigger and smaller primaries are defined as r1=(x+μ)2+y2 {r_1} = \sqrt {{{(x + \mu)}^2} + {y^2}} and r2=(x+μ1)2+y2 {r_2} = \sqrt {{{(x + \mu - 1)}^2} + {y^2}} .

Fig. 1

Configuration of the restricted three body problem.

Now, multiplying Eq. (1) by and Eq. (2) by , respectively and then adding them, we get x˙x¨+y˙y¨=Ωxx˙+Ωyy˙. \dot x{\kern 1pt} \ddot x + \dot y{\kern 1pt} \ddot y = {\Omega _x}{\kern 1pt} \dot x + {\Omega _y}{\kern 1pt} \dot y.

On integration, we obtain x˙2+y˙22=Ω(x,y)tCx˙2+y˙2=2ΩC, {{{{\dot x}^2} + {{\dot y}^2}} \over 2} = \int {{\partial {\kern 1pt} \Omega {\kern 1pt} (x,{\kern 1pt} y)} \over {\partial {\kern 1pt} t}} - C \Rightarrow {\dot x^2} + {\dot y^2} = 2{\kern 1pt} \Omega - C, where, C is constant of integration. Eq. (4) represents a well-known integral called as Jacobi integral in which C is known as Jacobi constant.

Existence of the Equilibrium Points

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 Ωx = 0, Ωy = 0, simultaneously i.e. Ωx=n2x(1μ)(x+μ)r13(1+3A212r12)μ(x+μ1)r23(1+3A222r22)=0, {\Omega _x} = {n^2}{\kern 1pt} x - {{(1 - \mu)(x + \mu)} \over {r_1^3}}\left({1 + {{3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} r_1^2}}} \right) - {{\mu (x + \mu - 1)} \over {r_2^3}}\left({1 + {{3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} r_2^2}}} \right) = 0, Ωy=n2y(1μ)yr13(1+3A212r12)μyr22(1+3A222r22)=0, {\Omega _y} = {n^2}{\kern 1pt} y - {{(1 - \mu){\kern 1pt} y} \over {r_1^3}}\left({1 + {{3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} r_1^2}}} \right) - {{\mu {\kern 1pt} y} \over {r_2^2}}\left({1 + {{3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} r_2^2}}} \right) = 0, for space coordinates x and y.

Collinear Equilibrium Points L1, L2, L3

Collinear equilibrium points are those points, which lie on the line joining the primaries. Therefore, for collinear equilibrium points y = 0 and hence r1 = |x + μ|, r2 = |x + μ − 1| so, Eq. (5) can be written as n2x(1μ)(x+μ)|x+μ|3(1+3A212(x+μ)2)μ(x+μ1)|x+μ1|3(1+3A222(x+μ1)2)=0. {n^2}{\kern 1pt} x - {{(1 - \mu)(x + \mu)} \over {|x + \mu {|^3}}}\left({1 + {{3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} {{(x + \mu)}^2}}}} \right) - {{\mu (x + \mu - 1)} \over {|x + \mu - 1{|^3}}}\left({1 + {{3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} {{(x + \mu - 1)}^2}}}} \right) = 0.

In the case of collinear equilibrium points, Eq. (6) is unimportant due to y = 0.

For simplicity, collinear axis is divided into three parts as (i)μ < x < 1 − μ; (ii) 1 − μ < x < +∞ and (iii) −∞< x < −μ, which correspond to the possible region of collinear equilibrium points L1, L2 and L3, respectively.

In case of L1 point (i.e for region −μ < x < 1 − μ), x + μ > 0, x + μ − 1 < 0 and hence, |x + μ| = x + μ, |x + μ − 1| = −(x + μ − 1). Therefore Eq. (7) reduces to n2x(1μ)|x+μ|2(1+3A212(x+μ)2)+μ|x+μ1|2(1+3A222(x+μ1)2)=0. {n^2}{\kern 1pt} x - {{(1 - \mu)} \over {|x + \mu {|^2}}}\left({1 + {{3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} {{(x + \mu)}^2}}}} \right) + {\mu \over {|x + \mu - 1{|^2}}}\left({1 + {{3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} {{(x + \mu - 1)}^2}}}} \right) = 0.

Similarly, in case of L2: x + μ > 0, x + μ − 1 > 0 and for L3: x + μ < 0, x + μ − 1 < 0, Eq. (7) is changed to n2x(1μ)|x+μ|2(1+3A212(x+μ)2)μ|x+μ1|2(1+3A222(x+μ1)2)=0, {n^2}{\kern 1pt} x - {{(1 - \mu)} \over {|x + \mu {|^2}}}\left({1 + {{3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} {{(x + \mu)}^2}}}} \right) - {\mu \over {|x + \mu - 1{|^2}}}\left({1 + {{3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} {{(x + \mu - 1)}^2}}}} \right) = 0, n2x+(1μ)|x+μ|2(1+3A212(x+μ)2)+μ|x+μ1|2(1+3A222(x+μ1)2)=0, {n^2}{\kern 1pt} x + {{(1 - \mu)} \over {|x + \mu {|^2}}}\left({1 + {{3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} {{(x + \mu)}^2}}}} \right) + {\mu \over {|x + \mu - 1{|^2}}}\left({1 + {{3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} {{(x + \mu - 1)}^2}}}} \right) = 0, respectively. The real solutions of Eqs (8)(10) lying in the respective regions will be the position of L1, L2 and L3, respectively. We have computed the collinear equilibrium points L1, L2 and L3 by varying the oblateness coefficient A21 and A22 in Jupiter-Europa system as shown in Table 1. From Table 1 it is observed that on increment in the oblateness coefficient A21 of the Jupiter shows more variation in the position of L1, but a small effect is seen both in case of L2 and L3 points. Due to the oblateness coefficient A22 of Europa, we found a considerable change in the position of L1, L2 and L3 points. An increment in the value of A22 corresponds to a shift in the position of L1 and L2 points towards Europa, but L3 slightly move away from the Jupiter. Further, points L1 and L3 move towards Jupiter on increment in the oblateness A22 of Europa while L2 moves away from the Europa.

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

A21 A22 L1 L2 L3 L4,5 : (x, ±y)

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

Triangular Equilibrium Points L4, L5

Since, for triangular equilibrium points y ≠ 0, therefore, these points are obtained by solving the Eqs (5) and (6), simultaneously. As, we know that in absence of all perturbations, r1 = r2 = 1 which implies x=12μ x = {1 \over 2} - \mu and y=±32 y = \pm {\kern 1pt} {{\sqrt 3} \over 2} [25]. Now, due to the presence of perturbation in the form of oblateness of the primaries, suppose r1 = 1 + ɛ1 and r2 = 1 + ɛ2, where ɛ1, ɛ2 are very small real numbers. Substituting the value of r1 and r2 in Eqs (5) and (6) then, solving for x and y by neglecting the second and higher order terms of ɛ1 and ɛ2 and their product, we get x=12(1+2ε12ε22μ), x = {1 \over 2}(1 + 2{\kern 1pt} {\varepsilon _1} - 2{\varepsilon _2} - 2{\kern 1pt} \mu), y=±123+4ε1+4ε2. y = \pm {1 \over 2}\sqrt {3 + 4{\kern 1pt} {\varepsilon _1} + 4{\kern 1pt} {\varepsilon _2}}.

The expression for ɛ1 and ɛ2 are obtained by solving the following system of equations a1ε1+a2ε2+a3=0, {a_1}{\kern 1pt} {\varepsilon _1} + {a_2}{\kern 1pt} {\varepsilon _2} + {a_3} = 0, b1ε1+b2ε2+b3=0. {b_1}{\kern 1pt} {\varepsilon _1} + {b_2}{\kern 1pt} {\varepsilon _2} + {b_3} = 0.

The system of Eqs (13) and (14) are achieved by putting the value of x and y from Eqs (11) and (12) into Eqs (5) and (6). And then simplifying resulting equations up to linear order terms of ɛ1 and ɛ2, where a1=n2+94(1μ)A2132μ(1A22),a2=1n2+32(1μ)A2132μ(1+3A222),a3=n22(12μ)+μ12+34(μ1)A21+34μA22,b1=14+39A21+4n23μ(6+13A21+2A22)43,b2=4+4n2+6A21(μ1)+μ(18+39A22)43,b3=34(2n22+3A21(μ1)3μA22). \matrix{{{a_1} = {n^2} + {9 \over 4}(1 - \mu){\kern 1pt} {A_{21}} - {3 \over 2}{\kern 1pt} \mu (1 - {A_{22}}),} \hfill \cr {{a_2} = 1 - {n^2} + {3 \over 2}(1 - \mu){\kern 1pt} {A_{21}} - {3 \over 2}{\kern 1pt} \mu \left({1 + {{3{\kern 1pt} {A_{22}}} \over 2}} \right),} \hfill \cr {{a_3} = {{{n^2}} \over 2}(1 - 2{\kern 1pt} \mu) + \mu - {1 \over 2} + {3 \over 4}(\mu - 1){\kern 1pt} {A_{21}} + {3 \over 4}{\kern 1pt} \mu {\kern 1pt} {A_{22}},} \hfill \cr {{b_1} = {{14 + 39{\kern 1pt} {A_{21}} + 4{\kern 1pt} {n^2} - 3{\kern 1pt} \mu (6 + 13{\kern 1pt} {A_{21}} + 2{\kern 1pt} {A_{22}})} \over {4\sqrt 3}},} \hfill \cr {{b_2} = {{- 4 + 4{\kern 1pt} {n^2} + 6{\kern 1pt} {A_{21}}(\mu - 1) + \mu {\kern 1pt} (18 + 39{\kern 1pt} {A_{22}})} \over {4\sqrt 3}},} \hfill \cr {{b_3} = {{\sqrt 3} \over 4}(2{\kern 1pt} {n^2} - 2 + 3{\kern 1pt} {A_{21}}{\kern 1pt} (\mu - 1) - 3{\kern 1pt} \mu {A_{22}}).} \hfill \cr}

Thus, substituting the value of ɛ1 and ɛ2 in Eqs (11) and (12) we get triangular equilibrium points L4; L5. In Table 1, apart from collinear equilibrium points, we have also presented the positions of triangular equilibrium points L4 and L5 at different values of oblateness coefficient A21 and A22 of the primaries. From Table 1, it is observed that the effect of the oblateness coefficient A21 of the Jupiter causes large variation in the positions of L4,5. However the oblateness coefficient A22 of Europa influence more in the positions of L4 and L5 points compared to that of A22. A combined effect of the approximate value of oblateness coefficients A21, A22 on equilibrium points L1,L2, L3, L4 and L5 have also been obtained as shown in bottom row of Table 1.

Linear Stability

Let the co-ordinate of the equilibrium point be denoted by (x0,y0). Suppose α and β denote small displacements of the infinitesimal mass from the equilibrium point (x0,y0), then position of infinitesimal mass at any future time is given as x = x0 + α, y = y0 + β. We assume that the displacements α and β are sufficiently small so that the Taylor's expansions of Ωx and Ωy about the equilibrium point (x0, y0) up to linear order terms in α, β, can be written as Ωx=αΩxx0+βΩxy0,Ωy=αΩyx0+βΩyy0. \matrix{{{\Omega _x} = \alpha {\kern 1pt} \Omega _{xx}^0 + \beta {\kern 1pt} \Omega _{xy}^0,} \hfill \cr {{\Omega _y} = \alpha {\kern 1pt} \Omega _{yx}^0 + \beta {\kern 1pt} \Omega _{yy}^0.} \hfill \cr}

Therefore, the equations of motion (1) and (2) in the neighbourhood of (x0,y0) can be written by using Taylor series expansion in the linear order as α¨2nβ˙=αΩxx0+βΩxy0, \ddot \alpha - 2n\dot \beta = \alpha {\kern 1pt} \Omega _{xx}^0 + \beta {\kern 1pt} \Omega _{xy}^0, β¨+2nα˙=αΩyx0+βΩyy0, \ddot \beta + 2n\dot \alpha = \alpha {\kern 1pt} \Omega _{yx}^0 + \beta {\kern 1pt} \Omega _{yy}^0, where Ωxx0 \Omega _{xx}^0 , Ωxy0 \Omega _{xy}^0 and Ωyy0 \Omega _{yy}^0 are second order partial derivatives at equilibrium point (x0, y0) and the expressions of Ωxx, Ωxy, Ωyy are given by Ωxx=n2+(1μ)r13[6(x+μ)23A212r12+15A21(x+μ)22r141]+μr23[6(x+μ+1)23A222r22+15A22(x+μ+1)22r241],Ωyy=n2+(1μ)r13[6y23A212r12+15A21y22r141]+μr23[6y23A222r22+15A22y22r241],Ωxy=(1μ)r15[3y(x+μ)+15A21(x+μ)y2r12]+μr25[15A22(x+μ1)y2r22+3(x+μ1)y]. \matrix{{{\Omega _{xx}} = {n^2} + {{(1 - \mu)} \over {r_1^3}}\left[ {{{6{\kern 1pt} {{(x + \mu)}^2} - 3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} r_1^2}} + {{15{\kern 1pt} {A_{21}}{\kern 1pt} {{(x + \mu)}^2}} \over {2{\kern 1pt} r_1^4}} - 1} \right]} \cr {+ {\mu \over {r_2^3}}\left[ {{{6{\kern 1pt} {{(x + \mu + 1)}^2} - 3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} r_2^2}} + {{15{\kern 1pt} {A_{22}}{{(x + \mu + 1)}^2}} \over {2{\kern 1pt} r_2^4}} - 1} \right],} \cr {{\Omega _{yy}} = {n^2} + {{(1 - \mu)} \over {r_1^3}}\left[ {{{6{\kern 1pt} {y^2} - 3{\kern 1pt} {A_{21}}} \over {2{\kern 1pt} r_1^2}} + {{15{\kern 1pt} {A_{21}}{\kern 1pt} {y^2}} \over {2{\kern 1pt} r_1^4}} - 1} \right] + {\mu \over {r_2^3}}\left[ {{{6{\kern 1pt} {y^2} - 3{\kern 1pt} {A_{22}}} \over {2{\kern 1pt} r_2^2}} + {{15{\kern 1pt} {A_{22}}{\kern 1pt} {y^2}} \over {2{\kern 1pt} r_2^4}} - 1} \right],} \cr {{\Omega _{xy}} = {{(1 - \mu)} \over {r_1^5}}\left[ {3{\kern 1pt} y(x + \mu) + {{15{\kern 1pt} {A_{21}}{\kern 1pt} (x + \mu){\kern 1pt} y} \over {2{\kern 1pt} r_1^2}}} \right] + {\mu \over {r_2^5}}\left[ {{{15{\kern 1pt} {A_{22}}{\kern 1pt} (x + \mu - 1){\kern 1pt} y} \over {2{\kern 1pt} r_2^2}} + 3{\kern 1pt} (x + \mu - 1){\kern 1pt} y} \right].} \cr}

Now, we investigate the motion of infinitesimal body about the equilibrium point (x0, y0) in xy-plane, for this we suppose α = Aeλ t, β = Beλ t where A, B, λ are parameters to be determined. Thus Eqs (15) and (16) reduces to A(λ2Ωxx0)+B(2nλΩxy0)=0A(2nλΩyx0)+B(λ2Ωyy0)=0. \matrix{{A{\kern 1pt} ({\lambda ^2} - \Omega _{xx}^0) + B{\kern 1pt} (- 2n\lambda - \Omega _{xy}^0) = 0} \hfill \cr {A{\kern 1pt} (2n\lambda - \Omega _{yx}^0) + B{\kern 1pt} ({\lambda ^2} - \Omega _{yy}^0) = 0.} \hfill \cr}

The above equations have a non-trivial solution for A and B if |λ2Ωxx02nλΩxy02nλΩyx0λ2Ωyy0|=0, \left| {\matrix{{{\lambda ^2} - \Omega _{xx}^0} & {- 2{\kern 1pt} n{\kern 1pt} \lambda - \Omega _{xy}^0} \cr {2{\kern 1pt} n{\kern 1pt} \lambda - \Omega _{yx}^0} & {{\lambda ^2} - \Omega _{yy}^0} \cr}} \right| = 0, which gives a bi-quadratic equation in λ known as characteristic equation λ4+P1λ2+Q1=0, {\lambda ^4} + {P_1}{\kern 1pt} {\lambda ^2} + {Q_1} = 0, where P1=4n2Ωxx0Ωyy0 {P_1} = 4{\kern 1pt} {n^2} - \Omega _{xx}^0 - \Omega _{yy}^0 and Q1=Ωxx0Ωyy0(Ωxy0)2 {Q_1} = \Omega _{xx}^0\Omega _{yy}^0 - {(\Omega _{xy}^0)^2} . The characteristic roots of characteristic Eq (17) are given as λ=±P1±P124Q12, \lambda = \pm \sqrt {{{- {P_1} \pm \sqrt {P_1^2 - 4{Q_1}}} \over 2}}, which are function of μ; A21 and A22. For the linear stability of equilibrium points, the nature of roots play a key role, which depends on the sign of discriminant P124Q1 P_1^2 - 4{\kern 1pt} {Q_1} of Eq. (17). Hence, in general, the solution of Eqs (15) and (16) are expressed as α(t)=A1eλ1t+A2eλ2t+A3eλ3t+A4eλ4t, \alpha (t) = {A_1}{e^{{\lambda _1}{\kern 1pt} t}} + {A_2}{e^{{\lambda _2}{\kern 1pt} t}} + {A_3}{e^{{\lambda _3}{\kern 1pt} t}} + {A_4}{e^{{\lambda _4}{\kern 1pt} t}}, β(t)=B1eλ1t+B2eλ2t+B3eλ3t+B4eλ4t, \beta (t) = {B_1}{e^{{\lambda _1}{\kern 1pt} t}} + {B_2}{e^{{\lambda _2}{\kern 1pt} t}} + {B_3}{e^{{\lambda _3}{\kern 1pt} t}} + {B_4}{e^{{\lambda _4}{\kern 1pt} t}}, where coefficients Bj are related to Aj by the relation as Bj=(λj2Ωxx02nλj)Aj;j=1,2,3,4. {B_j} = \left({{{\lambda _j^2 - \Omega _{xx}^0} \over {2{\kern 1pt} n{\kern 1pt} {\lambda _j}}}} \right){A_j}{\kern 1pt} ;\quad j = 1,2,3,4.

Linear Stability of Collinear Equilibrium Points

For the collinear equilibrium points y = 0 and hence, Ωxy0=Ωyx0=0 \Omega _{xy}^0 = \Omega _{yx}^0 = 0 . Therefore, in this case, the roots of characteristic Eq. (17) are of the form λ1=a,λ2=a,λ3=bi,λ4=bi. {\lambda _1} = a,\quad {\lambda _2} = - a,\quad {\lambda _3} = b{\kern 1pt} i,\quad {\lambda _4} = - b{\kern 1pt} i.

Thus, solutions of Eqs (15) and (16) can be written as α(t)=A1eat+A2eat+A3eibt+A4eibt, \alpha (t) = {A_1}{e^{a{\kern 1pt} t}} + {A_2}{e^{- a{\kern 1pt} t}} + {A_3}{e^{- i{\kern 1pt} b{\kern 1pt} t}} + {A_4}{e^{- i{\kern 1pt} b{\kern 1pt} t}}, β(t)=B1eat+B2eat+B3eibt+B4eibt. \beta (t) = {B_1}{e^{a{\kern 1pt} t}} + {B_2}{e^{- a{\kern 1pt} t}} + {B_3}{e^{- i{\kern 1pt} b{\kern 1pt} t}} + {B_4}{e^{- i{\kern 1pt} b{\kern 1pt} t}}.

The exponential factors eat and eat in Eqs (21) and (22) results in unbounded motion in the xy-plane, therefore collinear equilibrium points L1, L2 and L3 remain unstable for all 0 < μ ≤ 1/2.

Linear Stability of Triangular Equilibrium Points

In case of triangular equilibrium points, y ≠ 0. Therefore, the critical value of mass ratio can be obtained by solving the discriminant D=P124Q1 D = P_1^2 - 4{\kern 1pt} {Q_1} by equating it to zero, i.e. [2n2(1μ)(2r13+6A21r15)+μ(2r23+6A22r25)]2[1+2n2+(1μ)(2r13+6A21r15)+μ(2r23+6A22r25)]=0. \matrix{{{{\left[ {2{n^2} - (1 - \mu)\left({{2 \over {r_1^3}} + {{6{\kern 1pt} {A_{21}}} \over {r_1^5}}} \right) + \mu \left({{2 \over {r_2^3}} + {{6{\kern 1pt} {A_{22}}} \over {r_2^5}}} \right)} \right]}^2} -} \hfill \cr {\left[ {1 + 2{\kern 1pt} {n^2} + (1 - \mu)\left({{2 \over {r_1^3}} + {{6{\kern 1pt} {A_{21}}} \over {r_1^5}}) + \mu ({2 \over {r_2^3}} + {{6{\kern 1pt} {A_{22}}} \over {r_2^5}}} \right)} \right] = 0.} \hfill \cr}

In absence of the oblateness, i.e. when A21 = 0, A22 = 0, the Eq. (23) gives the critical mass ratio μc = 0.038529... [25], whereas in presence of oblateness (in particular at A21 = 0.000285, A22 = 0.007189), the critical value of mass ratio becomes μc = 0.024758. Thus, there arise three possible case as follows:

Case 1. When μc<μ<12 {\mu _c} < \mu < {1 \over 2} ; i.e. P2 − 4Q < 0

Let D=+P24Q D = + \sqrt {{P^2} - 4Q} , then Eq. (18) infer that four roots of the form a + ib. Due to positive real part, solutions (19) and (20) will contain exponential term and hence, it is clear that these type of roots make the triangular points instable.

Case 2. When μ = μc; i.e. P2 − 4Q = 0

This case implies P124Q1=0 P_1^2 - 4{\kern 1pt} {Q_1} = 0 , thus Eq. (18) yields λ1,2=iP12,λ3,4=iP12 {\lambda _{1,2}} = - i{\kern 1pt} \sqrt {{{{P_1}} \over 2}},\quad {\lambda _{3,4}} = i{\kern 1pt} \sqrt {{{{P_1}} \over 2}} which show that the roots are multiples of pure imaginary and hence, there appears secular terms in the solution. Therefore, the triangular points are unstable in this case too.

Case 3. When 0 < μ < μc; i.e. P2 − 4Q > 0

In this case, we found that the roots are of the form λ1,2=±iω1,λ3,4=±iω2, {\lambda _{1,2}} = \pm i{\kern 1pt} {\omega _1},\quad {\lambda _{3,4}} = \pm i{\kern 1pt} {\omega _2}, where ω1,2=P1±P124Q2, {\omega _{1,2}} = \sqrt {{{- {P_1} \pm \sqrt {P_1^2 - 4{\kern 1pt} Q}} \over 2}}, are real numbers. Thus, the resulting motion will be a combination of two periodic motions. Motion corresponds to the period T=2π|λ1,2|=2πω1 T = {{2{\kern 1pt} \pi} \over {|{\lambda _{1,2}}|}} = {{2{\kern 1pt} \pi} \over {{\omega _1}}} is known as short periodic motion and that corresponds to period T=2π|λ3,4|=2πω2 T = {{2{\kern 1pt} \pi} \over {|{\lambda _{3,4}}|}} = {{2{\kern 1pt} \pi} \over {{\omega _2}}} is called long periodic motion.

Further, we found that all characteristic roots at A21 = 0.000285, A22 = 0.007198 are purely imaginary for mass ratio 0 < μ < 0.0247585 (Figure 2) which indicates that L4,5 are stable there. Also, we have observed a variation in the nature of characteristic roots with respect to oblateness A21, A22 as given in Figure 3 and noticed that the oblateness of Europa affects more than that of oblateness of Jupiter on the stability region.

Fig. 2

Characteristic roots at A21 = 0.000285, A22 = 0.007198 in Jupiter-Europa system. Dashed lines represents imaginary roots and bold lines represents real roots.

Fig. 3

Characteristic roots for μ = 0.01 in Jupiter-Europa system at (a) A21 = 0.18436 and (b) A22 = 0.05459. Dashed lines represents imaginary roots and bold lines represents real roots.

Suppose, the coefficients Aj and Bj in solutions (19) and (20) are of the form Aj = aj + ibj and Bj = cj + idj where j = 1,2,3,4. For the motion of infinitesimal body, the displacement components α, β and the velocity components α˙ \dot \alpha , β˙ \dot \beta must be real. Since, the coefficients of exponential terms in solutions (19) and (20) exist in complex conjugate pairs [16] as follows Re(A1,A2)=a1,Re(A3,A4)=a2,Im(A1,A2)=b1,Im(A3,A4)=b2,Re(B1,B2)=c1,Re(B3,B4)=c2,Im(B1,B2)=d1,Im(A3,A4)=d2. \matrix{{Re({A_1},{A_2}) = {a_1},\quad Re({A_3},{A_4}) = {a_2},} \cr {Im({A_1}, - {A_2}) = {b_1},\quad Im({A_3}, - {A_4}) = {b_2},} \cr {Re({B_1},{B_2}) = {c_1},\quad Re({B_3},{B_4}) = {c_2},} \cr {Im({B_1}, - {B_2}) = {d_1},\quad Im({A_3}, - {A_4}) = {d_2}.} \cr}

Therefore, after simplification solutions (19) and (20) reduces to α(t)=2a1cosω1t+2a2cosω2t2b1sinω1t2b2sinω2t, \alpha (t) = 2{\kern 1pt} {a_1}{\kern 1pt} \cos {\omega _1}{\kern 1pt} t + 2{\kern 1pt} {a_2}{\kern 1pt} \cos {\omega _2}{\kern 1pt} t - 2{\kern 1pt} {b_1}{\kern 1pt} \sin {\omega _1}{\kern 1pt} t - 2{\kern 1pt} {b_2}{\kern 1pt} \sin {\omega _2}{\kern 1pt} t, β(t)=2c1cosω1t+2c2cosω2t2d1sinω1t2d2sinω2t. \beta (t) = 2{\kern 1pt} {c_1}{\kern 1pt} \cos {\omega _1}{\kern 1pt} t + 2{\kern 1pt} {c_2}{\kern 1pt} \cos {\omega _2}{\kern 1pt} t - 2{\kern 1pt} {d_1}{\kern 1pt} \sin {\omega _1}{\kern 1pt} t - 2{\kern 1pt} {d_2}{\kern 1pt} \sin {\omega _2}{\kern 1pt} t.

In particular at L4 (0.4965504, 0.8638773), μ = 0.01, A21 = 0.000285 and A22 = 0.007198, the perturbed solution α and β with initial conditions α(0) = 10−4, β (0) = 10−4 and α˙(0)=0 \dot \alpha (0) = 0 , β˙(0)=0 \dot \beta (0) = 0 , are expressed as α(t)2×104(1.91cosω1t1.41cosω2t35.96sinω1t19.45sinω2t), \alpha (t) \approx 2 \times {10^{- 4}}{\kern 1pt} \left({1.91{\kern 1pt} \cos {\omega _1}{\kern 1pt} t - 1.41{\kern 1pt} \cos {\omega _2}{\kern 1pt} t - 35.96{\kern 1pt} \sin {\omega _1}{\kern 1pt} t - 19.45{\kern 1pt} \sin {\omega _2}{\kern 1pt} t} \right), β(t)2×104(11.89cosω1t11.24cosω2t18.84sinω1t8.78sinω2t), \beta (t) \approx 2 \times {10^{- 4}}{\kern 1pt} \left({- 11.89{\kern 1pt} \cos {\omega _1}{\kern 1pt} t - 11.24{\kern 1pt} \cos {\omega _2}{\kern 1pt} t - 18.84{\kern 1pt} \sin {\omega _1}{\kern 1pt} t - 8.78{\kern 1pt} \sin {\omega _2}{\kern 1pt} t} \right), where, ω1,2 ≈ ±0.355 and ω3,4 ≈ ±0.948. The bounded and periodic nature of solutions (26) and (27) infer that motion in the neighbourhood of L4,5 are stable. The semi analytical solution α(t) and β (t) graphically can be seen in Figure 4.

Fig. 4

The perturbed solution α(t), β (t) about the L4 point at A21 = 0.000285 and A22 = 0.007198 for time period T ≈ 57.12.

Tadpole and horseshoe orbits about triangular equilibrium points

In the RTBP, there exist different types of motion of an infinitesimal mass in the vicinity of the L4,5, 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 L4,5. He has found two set of initial conditions for horseshoe orbits and for long tadpole-shaped orbits around the L4,5. Also, Taylor [26] briefly has described the smooth horseshoe orbits around L4,5 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 P(x, y) lie on the straight line joining the Jupiter (bigger primary) and L4 or L5, which is the closest approach for P towards L4 or L5. A good approximation for periodic orbit about L4,5 is that the magnitude of velocity V of the infinitesimal mass changes slowly in such a way that Vt {{\partial V} \over {\partial t}} and 2V2t {{{\partial ^2}V} \over {{\partial ^2}t}} approaches to zero. Let the distance from the infinitesimal body P to the point L4 be k at any instant, then the coordinates of infinitesimal mass in the rotating frame are x=(1+k)2μ x = {{(1 + k)} \over 2} - \mu and y=3(1+k)2 y = {{\sqrt 3 (1 + k)} \over 2} . For different value of k, we generate the set of initial conditions. Thus, integrating the equations of motion (1) and (2) in rotating frame with these initial conditions under the presence of oblateness of primaries in Jupiter-Europa system, we obtain tadpole and horseshoe orbits. Numerical results for different values of k are shown in Figures 5 and 6, in which we have acquired tadpole and horseshoe orbits around the L4 and L5 in the Jupiter-Europa system in presence and absence of oblateness of Jupiter and Europa. For the initial conditions corresponding to the value of k = 0.0001, we have found a narrow tadpole orbits around L4 (also, similar tadpole orbits about L5 are obtained which are not presented here) at μ = 0.0000251 and A21 = A22 = 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 (A21 = 0.0285), but a slight increment in the oblateness of Europa (A22 = 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 L4 and L5 at k = 0.005 are displayed. From Figure 6(a–d), it is noticed that in the presence of oblateness, the number of loops in the horseshoe orbits get varied. Thus, due to oblateness of the primaries, the shape and size of the tadpole and horseshoe orbits in the Jupiter-Europa system deviated from the classical one [16, 26].

Fig. 5

Tadpole orbit in Jupiter (J) - Europa (E) system for k = 0.0001 at (a) A21 = 0 = A22 (b) A21 = 0.0285, A22 = 0 (c) A21 = 0, A22 = 0.0007189 and (d) A21 = 0.0285, A22 = 0.0007189.

Fig. 6

Horseshoe orbit in Jupiter (J) - Europa (E) system for k = 0.005 at (a) A21 = 0 = A22 (b) A21 = 0.00285, A22 = 0 (c) A21 = 0, A22 = 0.0007189 and (d) A21 = 0.00285, A22 = 0.0007189.

Poincaré surfaces of section (PSS)

The position coordinate (x, y) and corresponding velocity (, ) at any instant corresponds to a point in a four-dimensional phase space, which can be reduced to three-dimensional phase space due to the existence of Jacobi constant C Eq. (4). In order to compute surfaces of section, we fix the plane y = 0 in the three dimensional space to get the values of x and which can be plotted with respect to time whenever the trajectory crosses the plane y = 0. This technique is called PSS or Poincaré map [20]. The information, whether the trajectory is regular or chaotic in nature, can be obtained with the help of PSS. Also, we can identify the periodic and quasi-periodic orbits in the phase space by using PSS technique. In the surface of section, the formation of chain of islands corresponds to a single quasi-periodic trajectory. The evolution of dusty regions between the curves and the chain of islands represents chaotic zones.

We have computed the PSS at different values of mass ratio μ, Jacobi constant Cj and oblateness coefficients A21, A22 by the method described in [7,16,27]. The starting value of x are taken in such a way that > 0 for each value of Cj with y = = 0. For the range of Jacobi constant Cj between 2.55 and 3.30, we have integrated the equations of motion with starting conditions where the distance between the two consecutive starting conditions is chosen as 0.01 at each Cj. Different surfaces of section are generated as shown in Figures 710. From Figure 7, we can see that the chaotic zones increases with the decrease in the mass ratio μ. Also, we have plotted the surface of section for Jupiter-Europa system for μ = 0.0000251 (Figure 7). For 2.55 < Cj < 3.04, it is observed that chain of islands corresponds to a quasi-periodic trajectory increases gradually and formation of the chaotic zones starts (Figure 8). Further, increasing the value of Cj, there remains chain of islands corresponding to quasi-periodic orbits, but reduces the chaotic regions. As, the oblateness of primaries A21 and A22 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 A22 than that of bigger primary A21. In presence of oblateness, it is found that the curves or chain of islands representing quasi-periodic orbits predominates the chaotic regions.

Fig. 7

Poincaré surfaces of section at different mass ratio μ.

Fig. 8

Variation in Poincaré surfaces of section for μ = 0.001 at different values of Cj.

Fig. 9

Poincaré surfaces of section for μ = 0.001, Cj = 2.85 with the variation in the oblateness A21.

Fig. 10

Effect of oblateness A22 on Poincaré surfaces of section with μ = 0.001, Cj = 2.85.

Conclusion

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 < μc < 0.038529... [25] to 0 < μc < 0.024758... however, collinear equilibrium points are unstable for 0 < μ ≤ 1/2 as in classical case [25]. Moreover, positions of equilibrium points L1,2,3,4,5 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 k, which are displayed in Figures 56. For the initial conditions corresponding to the value of k = 0.0001, we have found narrow tadpole orbits around L4 at = 0.0000251 and A21 = A22 = 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 (A21 = 0.0285), but a slight increment in the oblateness of Europa (A22 = 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 L4 and L5 at k = 0.005 (Figures 6), it is noticed that in the presence of oblateness, a small difference in the loops occurs (Figures 6b, 6c, 6d). Because of the oblateness of the primaries, the results on tadpole and horseshoe orbits of the model in hand, shows a deviation from the results of classical case [16, 26]. Further, we have described the nature of orbits in presence of oblateness by using Poincaré surfaces of sections technique and computed the surfaces of section at different values of, Cj, A21 and A22 (Figures 710). 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 A21 and A22 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.

Fig. 1

Configuration of the restricted three body problem.
Configuration of the restricted three body problem.

Fig. 2

Characteristic roots at A21 = 0.000285, A22 = 0.007198 in Jupiter-Europa system. Dashed lines represents imaginary roots and bold lines represents real roots.
Characteristic roots at A21 = 0.000285, A22 = 0.007198 in Jupiter-Europa system. Dashed lines represents imaginary roots and bold lines represents real roots.

Fig. 3

Characteristic roots for μ = 0.01 in Jupiter-Europa system at (a) A21 = 0.18436 and (b) A22 = 0.05459. Dashed lines represents imaginary roots and bold lines represents real roots.
Characteristic roots for μ = 0.01 in Jupiter-Europa system at (a) A21 = 0.18436 and (b) A22 = 0.05459. Dashed lines represents imaginary roots and bold lines represents real roots.

Fig. 4

The perturbed solution α(t), β (t) about the L4 point at A21 = 0.000285 and A22 = 0.007198 for time period T ≈ 57.12.
The perturbed solution α(t), β (t) about the L4 point at A21 = 0.000285 and A22 = 0.007198 for time period T ≈ 57.12.

Fig. 5

Tadpole orbit in Jupiter (J) - Europa (E) system for k = 0.0001 at (a) A21 = 0 = A22 (b) A21 = 0.0285, A22 = 0 (c) A21 = 0, A22 = 0.0007189 and (d) A21 = 0.0285, A22 = 0.0007189.
Tadpole orbit in Jupiter (J) - Europa (E) system for k = 0.0001 at (a) A21 = 0 = A22 (b) A21 = 0.0285, A22 = 0 (c) A21 = 0, A22 = 0.0007189 and (d) A21 = 0.0285, A22 = 0.0007189.

Fig. 6

Horseshoe orbit in Jupiter (J) - Europa (E) system for k = 0.005 at (a) A21 = 0 = A22 (b) A21 = 0.00285, A22 = 0 (c) A21 = 0, A22 = 0.0007189 and (d) A21 = 0.00285, A22 = 0.0007189.
Horseshoe orbit in Jupiter (J) - Europa (E) system for k = 0.005 at (a) A21 = 0 = A22 (b) A21 = 0.00285, A22 = 0 (c) A21 = 0, A22 = 0.0007189 and (d) A21 = 0.00285, A22 = 0.0007189.

Fig. 7

Poincaré surfaces of section at different mass ratio μ.
Poincaré surfaces of section at different mass ratio μ.

Fig. 8

Variation in Poincaré surfaces of section for μ = 0.001 at different values of Cj.
Variation in Poincaré surfaces of section for μ = 0.001 at different values of Cj.

Fig. 9

Poincaré surfaces of section for μ = 0.001, Cj = 2.85 with the variation in the oblateness A21.
Poincaré surfaces of section for μ = 0.001, Cj = 2.85 with the variation in the oblateness A21.

Fig. 10

Effect of oblateness A22 on Poincaré surfaces of section with μ = 0.001, Cj = 2.85.
Effect of oblateness A22 on Poincaré surfaces of section with μ = 0.001, Cj = 2.85.

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

A21 A22 L1 L2 L3 L4,5 : (x, ±y)

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*

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