Analytical and Semianalytical Treatment of the Collinear Points in the Photogravitational Relativistic RTBP

Abstract

The problem of restricted three bodies is considered. The treatment is carried out within the framework of the post-Newtonian approximation. The primaries also are assumed to be radiant sources. The locations of the collinear points are computed. Series forms of these locations are obtained as new results. A Mathematica program is constructed so as to draw the locations of collinear points versus the whole range of the mass ratio 𝜇 taking into account the photo-gravitational effects and/or the relativistic corrections. Analyses of these figures are addressed.

1. Introduction

The restricted three-body problem (in brief RTBP) is the most celebrated problem in celestial mechanics, wherein one is interested in the motion of a test particle, a body of negligible mass called infinitesimal mass in the presence of two massive bodies called primaries revolving around their common centre of mass. Attempts at its solution laid the foundation for dynamical systems theory and alerted Poincaré in the 1890s to the existence of deterministic chaos within Newtonian mechanics. There is no general analytical solution in terms of simple algebraic functions for the problem of three gravitating bodies of comparable masses. Except in a few very specific cases the problem has to be solved numerically. Are there any points where an infinitesimal mass could orbit around the centre of mass with the same period as the primaries? Euler found three such points, the collinear points, on the line joining the primaries. Lagrange found two additional points, the fourth and fifth triangular points. They are not collinear with 𝑚₁ and 𝑚₂ but are such that the three masses are at the corners of an equilateral triangle. The five points are called the equilibrium points.

These locations of the equilibrium points can be found by setting the first derivatives of the potential to zero. Note that, in the line joining the two masses, the equivalent potential at the collinear points is a maximum, and therefore these points are unstable; these points are actually saddle points. In several spacecrafts which are in orbit or are planned to be in orbit around the collinear points (e.g., SOHO at the interior collinear point and MAP at 𝐿₂), this instability has already used in consuming small expenditure of fuel which is presumably needed to keep them there.

The classical restricted three-body problem concerned with the determination of the locations of the libration points and investigating their linear/nonlinear stability. The RTBP becomes photo-gravitational when one or both of the masses of the primaries is an intense emitter of radiation, Kumar and Ishwar [1]. The restricted three body problem describes the dynamical system more accurately on account of realistic assumptions of the potential field and of the motion of the primaries, Narayan and Kumar [2].

The location and stability of the libration points in the problem of three bodies are subject to modifications when including one or more of additional forces in the potential of the classical problem as the radiation pressure component of the radiation drag which is the next most powerful component after the gravitational forces, Schuerman [3]; the oblateness of the primaries, Oberti and Vienne [4]; the primaries orbits being elliptic, Kumar and Ishwar [5]; the relativistic corrections to RTBP Abd El-Salam and Abd El-Bar [6], Douskos and Perdios [7], and Ragos et al. [8]; the aerodynamic drag, Markakis et al. [9]; Douskos [10] and the eternal radiation pressure as the effect of the direct solar radiation pressure on the equilibrium points in the Earth-Moon system, Poynting-Robertson (P-R) drag, and so forth.

The literature is wealthy of works on the subject of the restricted three body problem in classical form since its eminent pioneer Lagrange to the very recent date when considering different sources of perturbations and it will be worth noting to sketch some of these most important as well as recent works: Sharma [11], Hallan and Rana [12], Sándor and Érdi [13], Floría [14], Markeev [15], Palacián et al. [16, 17], Bruno and Varin [18], Ammar [19], Narayan et al. [20–22], Kushvah and [23–25], Douskos [26], Shankaran et al. [27].

In this work, we consider the collinear equilibrium points of the relativistic RTBP amended with the photo-gravitational effects of the two primaries. We determine new formulas for approximate positions of the collinear points as series expansions. Terms are ordered in 𝜇 and 1/𝑐² under the considered perturbations.

The rest of the paper is organized as follows. In Section 2 we formulated the equations of motion of the infinitesimal mass in the relativistic RTBP in a synodic frame of reference taking into account the radiation of the primaries. In Section 3 we computed the perturbed locations of the collinear libration points. In Section 4 we designed a Mathematica program to draw the locations of collinear points 𝐿_𝑖,𝑖=1,2,3, versus the whole range of the mass ratio 𝜇 taking into account the photogravitational effects and/or the relativistic corrections. In Section 5 we concluded our work and highlighted the differences between our work and previous works. Also we mentioned our forthcoming works for the sake of completeness.

2. Equations of Motion

The equations of motion of the infinitesimal mass in the relativistic RTBP in a synodic frame of reference (𝜉,𝜂), in which the primary coordinates on the 𝜉-axis (−𝜇,0),(1−𝜇,0) are kept fixed and the origin at the center of mass, are given by Brumberg [28] and Bhatnagar and Hallan [29]: 𝜉̈−2𝑛𝜂̇=𝜕𝑈−𝑑𝜕𝜉𝑑𝑡𝜕𝑈𝜕𝜉̇,𝜂̈+2𝑛𝜉̇=𝜕𝑈−𝑑𝜕𝜂𝑑𝑡𝜕𝑈𝜕𝜂̇ (1) with 1𝑛=1+2𝑐²𝜇1−𝜇−3,𝑟=𝜉²+𝜂², (2) where 𝑈 is the potential-like function of the relativistic RTBP, which can be written as composed of two components 𝑈_𝑐+phand 𝑈_𝑟+ph, namely, on respective the potential of the classical RTBP gravitational potential and the relativistic terms corrected for the radiation of the primaries which lead to mass reduction: 𝑈=𝑈_𝑐+ph+𝑈_𝑟+ph, (3) where 𝑈_𝑐+phand 𝑈_𝑟+ph are given by 𝑈_𝑐+ph=𝑟²2+𝑞₁1−𝜇𝑟₁+𝑞₂𝜇𝑟₂,𝑈_𝑟+ph=𝑟²2𝑐²𝜇+11−𝜇−38𝑐²×𝜉+𝜂̇²+𝜂−𝜉̇²²+32𝑐²𝑞₁1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝜉+𝜂̇²+𝜂−𝜉̇²−12𝑐²𝑞₁1−𝜇𝑟₁+𝑞₂𝜇𝑟₂²−𝑞₁𝑞₂𝜇1−𝜇2𝑐²×1𝑟₁+1𝑟₁−1𝑟₂1−3𝜇−7𝜉−8𝜂̇+𝜂²𝑞₂𝜇𝑟³₁+𝑞₁1−𝜇𝑟³₂,𝑟₁=𝜉+𝜇²+𝜂²,𝑟₂=𝜉+𝜇−1²+𝜂², (4) where 𝑞_𝑖=1−(𝐹_𝑝/𝐹_𝑔)_𝑖,𝑖=1,2, is the mass reduction factor constant for a given particle. 𝐹_𝑝, is the force due to radiation pressure, and 𝐹_𝑔 is the force due to gravitational field, Sharma et al. [30], Simmons et al. [31], and Chernikov [32] with a modification to account for the radiation of the primaries 𝑞_𝑖.

3. Location of the Libration Points

From the equations of motion (1), it is apparent that an equilibrium solution exists relative to the rotating frame when the partial derivatives of the pseudopotential function (𝑈_𝜉,𝑈_𝜂,𝑈_𝜁) are all zero; that is, 𝑈 = const. These points correspond to the positions in the rotating frame at which the gravitational forces and the centrifugal force associated with the rotation of the synodic frame all cancel, with the result that a particle located at one of these points appears stationary in the synodic frame. There are five equilibrium points in the circular RTBP, also known as Lagrange points or libration points. Three of them (the collinear points) lie along the 𝜉-axis: one interior point 𝐿₁ between the two primaries, and one point on the far side of each primary with respect to the barycenter 𝐿₂,𝐿₃. The other two libration points (the triangular points) are each located at the apex of an equilateral triangle formed with the primaries; see Figure 1. The triangular points are designated as 𝐿₄ and 𝐿₅, with 𝐿₄ moving ahead of the 𝜉-axis and 𝐿₅ trailing 𝜉-axis, along the orbit of the small primary as the synodic frame rotates uniformly relative to the inertial frame (as shown in Figure 1).

Figure 1 

The five Lagrangian points with Sun-Earth system as an example.

Remark 1. All the five libration points lie in the 𝜉-𝜂 plane, that is, in the plane of motion.

The libration points are obtained from equations of motion after setting 𝜉̈=𝜂̈=𝜉̇=𝜂̇=0.These points represent particular solutions of equations of motion 𝜕𝑈=𝜕𝜉𝜕𝑈_𝑐+ph+𝜕𝜉𝜕𝑈_𝑟+ph𝜕𝜉=0,𝜕𝑈=𝜕𝜂𝜕𝑈_𝑐+ph+𝜕𝜂𝜕𝑈_𝑟+ph𝜕𝜂=0. (5) The explicit required partial derivatives are (remembering that 𝑈=𝑈_𝑐+ph+𝑈_𝑟+ph) 𝜕𝑈𝑞𝜕𝜉=𝜉−¹1−𝜇𝜉+𝜇𝑟³₁−𝑞₂𝜇𝜉+𝜇−1𝑟³₂+1𝑐²×𝜇−𝜇²𝑞−3𝜉+¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝜉+𝜇𝑟³₁+𝑞₂𝜇𝜉+𝜇−1𝑟³₂+12𝜂²+𝜉²3𝜉−2×𝑞₁1−𝜇𝜉+𝜇𝑟³₁+𝑞₂𝜇𝜉+𝜇−1𝑟³₂×𝜂²+𝜉²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂𝜉−12𝑞₁𝑞₂𝜇×−11−𝜇𝑟³₁+𝜉+𝜇𝜉+𝜇𝑟³₁−𝜉+𝜇−1𝑟³₂×1−1+3𝜇+7𝜉−7𝑟₁−1𝑟₂−3𝜂²×𝑞₂𝜇𝜉+𝜇𝑟⁵₁+𝑞₁1−𝜇𝜉+𝜇−1𝑟⁵₂=0,𝜕𝑈𝑞𝜕𝜂=𝜂−¹1−𝜇𝑟³₁𝑞𝜂−²𝜇𝑟³₂𝜂+1𝑐²𝜇−𝜇²𝑞−3𝜂+¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝑟³₁+𝑞₂𝜇𝑟³₂1𝜂+2𝜉²+𝜂²𝜂−32𝜉²+𝜂²𝑞₁1−𝜇𝑟³₁+𝑞₂𝜇𝑟³₂𝜂𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂𝑞𝜂+¹𝑞₂𝜇1−𝜇2𝜂+12𝑞₁𝑞₂𝜇𝜂×−11−𝜇𝑟³₁+1𝑟³₂𝑞−1+3𝜇+7𝜉+3¹1−𝜇𝑟⁵₂+𝑞₂𝜇𝑟⁵₁𝜂²+1𝑟³₁𝑞−2²𝜇𝑟³₁+𝑞₁1−𝜇𝑟³₂=0. (6)(7)

4. Location of Collinear Libration Points

Any point of the collinear points must, by definition, have 𝜁=𝜂=0, and the solution of the classical RTBP satisfies 𝜎₁𝑟₁+𝜎₂𝑟₂𝑟=1,₁=𝜎₁,𝑟𝜉+𝜇₂=−𝜎₂𝜇+𝜉−1, (8) where (see Figures 2, 3, and 4) we have 𝐿₁∶𝜎₁=1,𝜎₂𝐿=1,₂∶𝜎₁=1,𝜎₂𝐿=−1,₃∶𝜎₁=−1,𝜎₂=1. (9)

Figure 2 

The Location of 𝐿1.

Figure 3 

The Location of 𝐿2.

Figure 4 

The Location of 𝐿3.

4.1. Location of 𝐿₁

Substituting from (8) with the corresponding values of 𝜎₁=1 and 𝜎₂=1 into (6) and noting that 𝜕𝑟₁/𝜕𝜉=−𝜕𝑟₂/𝜕𝜉=1, we get 𝜕𝑈𝑞𝜕𝜉=𝜉−¹1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂+1𝑐²×𝜇−𝜇²𝑞−3𝜉+¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂−32𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂𝜉²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂1𝜉+2𝜉³−12𝑞₁𝑞₂𝜇×−11−𝜇𝑟²₁+1𝑟²₁+1𝑟²₂1−1+3𝜇+7𝜉−7𝑟₁−1𝑟₂=0 (10) which can be re-written as a function of 𝑟₁,𝑟₂ as 𝜕𝑈𝜕𝜉=1−𝜇−𝑟₂−𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂+1𝑐²𝑞₁1−𝜇𝑟₁+𝑞₂𝜇𝑟₂𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂+𝜇−𝜇²−3×1−𝜇−𝑟₂+121−𝜇−𝑟₂³−32𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂×1−𝜇−𝑟₂²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×1−𝜇−𝑟₂−12𝑞₁𝑞₂𝜇×−11−𝜇𝑟²₁+1𝑟²₁+1𝑟²₂×−1+3𝜇+71−𝜇−𝑟₂1−7𝑟₁−1𝑟₂=0. (11) Then it may be reasonable in our case to assume that positions of the equilibrium points 𝐿₁ are the same as given by classical RTBP but perturbed due to the inclusion of the relativistic correction by quantities 𝜀₁≡𝒪(1/𝑐²). See Soffel [33] and Douskos and Perdios [7].

The solution of the classical part of (10) including the photogravitational correction can be obtained easily using the potential 𝑈_𝑐+ph=𝑞₁(1−𝜇)(𝑟²₁+2/𝑟₁)+𝑞₂𝜇(𝑟²₂+2/𝑟₂) as follows: 𝜕𝑈_𝑐+ph=𝜕𝜉𝜕𝑈_𝑐+ph𝜕𝑟₁𝜕𝑟+𝜕𝜉𝜕𝑈_𝑐+ph𝜕𝑟₂𝜕𝑟₂⟹𝜕𝜉=0,𝜕𝑈_𝑐+ph=𝜕𝜉𝜕𝑈_𝑐+ph𝜕𝑟₁−𝜕𝑈_𝑐+ph𝜕𝑟₂⟹=0𝜕𝑈_𝑐+ph𝜕𝑟₁=𝜕𝑈_𝑐+ph𝜕𝑟₂⟹𝑞₂𝜇𝑞₁=1−𝜇3𝑟₂−3𝑟²₂+𝑟³₂𝑟²₂1−𝑟₂²1−𝑟³₂. (12) If (𝑞₂𝜇)/(𝑞₁(1−𝜇)) is small, this equation has a root in the vicinity of 𝑟₂=𝛼, where 𝛼=((𝑞₂𝜇)/(3𝑞₁(1−𝜇)))^1/3; this 𝛼 can be written as 𝛼=𝑟₂11+3𝑟₂+13𝑟²₂+53𝑟81³²+101𝑟243⁴²+𝒪𝑟⁵₂+⋯. (13) Let successive approximations of 𝑟₂ be 𝑟_2𝑖: 𝑟₂₀=𝛼,𝛼=𝑟₂₁11+3𝛼,𝑟₂₁1=𝛼1−3𝛼,𝛼=𝑟₂₂11+32𝛼+9𝛼²,⋮+⋯ (14) Then successive approximation yields 𝑟₂₄𝛼=𝛼1−3−𝛼²9+2𝛼27³+2𝛼81⁴+𝒪(𝛼)⁵. (15) Let us set the following notation: 𝑟₁=𝑎₁+𝜀₁,𝑟₂=𝑏₁−𝜀₁,𝑎₁=1−𝑏₁, (16) where 𝑎₁ and 𝑏₁ are unperturbed relativistically positions of 𝑟₁ and 𝑟₂, respectively, and 𝑏₁ is given after some successive approximation by the relation 𝑏₁=𝑟₂₄𝛼=𝛼1−3−𝛼²9+2𝛼27³+2𝛼81⁴+𝒪(𝛼)⁵. (17) Substituting from (16) into (11) and retaining the terms up to the first order in the small quantities 𝜀₁, we can get 𝜕𝑈𝜕𝜉=1−𝜇−𝑏₁+𝜀₁−𝑞₁1−𝜇1−𝑏₁²1−2𝜀₁1−𝑏₁+𝑞₂𝜇𝑏²₁1+2𝜀₁𝑏₁+1𝑐²×𝜇−𝜇²−31−𝜇−𝑏₁+𝑞₁1−𝜇1−𝑏₁+𝑞₂𝜇𝑏₁𝑞₁1−𝜇1−𝑏₁²−𝑞₂𝜇𝑏²₁+121−𝜇−𝑏₁³−32×𝑞₁1−𝜇1−𝑏₁²−𝑞₂𝜇𝑏²₁×1−𝜇−𝑏₁²𝑞+3¹1−𝜇1−𝑏₁+𝑞₂𝜇𝑏₁×1−𝜇−𝑏₁−12𝑞₁𝑞₂𝜇×−11−𝜇1−𝑏₁²1−71−𝑏₁−1𝑏₁+11−𝑏₁²+1𝑏²₁×6−7𝑏₁−4𝜇=0. (18) Equation (18) can be solved for 𝜀₁; then substituting back into (16) and (8) yields, after some lengthy algebraic manipulations, the location of 𝐿₁ as 𝜉_0,𝐿1−=1−𝜇_5𝑘=0ℳ^𝐿1_𝑘+1𝑐²𝒩^𝐿1_𝑘𝜇3^𝑘/3𝜇+𝒪3², (19) where the nonvanishing coefficients ℳ^𝐿1_𝑘 and 𝒩^𝐿1_𝑘, are collected in Appendix A.

4.2. Location of 𝐿₂

The collinear point 𝐿₂ is located outside the small massive primary of mass 𝜇. We now drive an approximate location for this point under the considered perturbations. Substituting from (8) with the corresponding values of 𝜎₁=1 and 𝜎₂=−1 into (6) and noting that 𝜕𝑟₁/𝜕𝜉=𝜕𝑟₂/𝜕𝜉=1, we get 𝑟₁−𝑟₂𝑟=1,₁𝑟=𝜉+𝜇,₂=𝜉+𝜇−1,𝜕𝑟₁=𝜕𝜉𝜕𝑟₂𝜕𝜉=1. (20) Substituting from (20) into (6), we get 𝜕𝑈𝑞𝜕𝜉=𝜉−¹1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂+1𝑐²×𝜇−𝜇²𝑞−3𝜉+¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂−32𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂𝜉²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂1𝜉+2𝜉³−12𝑞₁𝑞₂𝜇×−11−𝜇𝑟²₁+1𝑟²₁−1𝑟²₂×1−1+3𝜇+7𝜉−7𝑟₁−1𝑟₂=0 (21) which can be rewritten as a function of 𝑟₁ and 𝑟₂ as 𝜕𝑈𝜕𝜉=1−𝜇+𝑟₂−𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂+1𝑐²𝜇−𝜇²𝑞−3𝜉+¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂−32𝑞₁1−𝜇𝑟²₁−𝑞₂𝜇𝑟²₂𝜉²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂1𝜉+2𝜉³−12𝑞₁𝑞₂𝜇×−11−𝜇𝑟²₁+1𝑟²₁−1𝑟²₂×1−1+3𝜇+7𝜉−7𝑟₁−1𝑟₂=0. (22) Then, it may be reasonable in our case to assume that the position of the equilibrium point 𝐿₂ is the same as given by classical RTBP but perturbed due to the inclusion of the relativistic correction by quantities 𝜀₂≡𝒪(1/𝑐²): 𝑟₁=𝑎₂+𝜀₂,𝑟₂=𝑏₂+𝜀₂,𝑎₂=1+𝑏₂ (23) with 𝑎₂ and 𝑏₂ being the unperturbed positions of 𝑟₁ and 𝑟₂, respectively, where 𝑏₂ is given after some successive approximation as 𝑏₂𝛼=𝛼1+3−𝛼²9−2𝛼27³+2𝛼81⁴+𝒪(𝛼)⁵,𝑞𝛼=²𝜇3𝑞₁(1−𝜇)^1/3. (24) Substituting from (23) into (22) and retaining the terms up to the first order in the small quantities 𝜀₂, we can get 𝜕𝑈𝜕𝜉=1−𝜇+𝑏₂+𝜀₂−𝑞₁1−𝜇1+𝑏₂²×1−2𝜀₂1+𝑏₂−𝑞₂𝜇𝑏²₂1−2𝜀₂𝑏₂+1𝑐²𝜇−𝜇²−31−𝜇+𝑏₂+𝑞₁1−𝜇1+𝑏₂+𝑞₂𝜇𝑏₂×𝑞₁1−𝜇1+𝑏₂²+𝑞₂𝜇𝑏²₂+121−𝜇+𝑏₂³−32𝑞₁1−𝜇1+𝑏₂²+𝑞₂𝜇𝑏²₂×1−𝜇+𝑏₂²×𝑞+3₁1−𝜇1+𝑏₂+𝑞₂𝜇𝑏₂×1−𝜇+𝑏₂−12𝑞₁𝑞₂𝜇×−11−𝜇1+𝑏₂²1−71+𝑏₂−1𝑏₂+11+𝑏₂²−1𝑏²₂6+7𝑏₂−4𝜇=0. (25) Equation (25) can be solved for 𝜀₂; then substituting back into (23) and (20) yields, after some lengthy algebraic manipulations, the location of 𝐿₂ as 𝜉_0,𝐿2+=1−𝜇_5𝑘=0ℳ^𝐿2_𝑘+1𝑐²𝒩^𝐿2_𝑘𝜇3^𝑘/3𝜇+𝒪3², (26) where the non-vanishing coefficients ℳ^𝐿2_𝑘 and 𝒩^𝐿2_𝑘, are collected in Appendix B.

4.3. Location of 𝐿₃

The 𝐿₃ point lies on the negative 𝜉-axis. We now derive an approximate location for this point. Substituting from (8) with the corresponding values of 𝜎₁=−1 and 𝜎₂=1 into (6) and noting that 𝜕𝑟₁/𝜕𝜉=𝜕𝑟₂/𝜕𝜉=−1, we get 𝑟₂−𝑟₁𝑟=1,₁,𝑟=−𝜉+𝜇₂=1−𝜇−𝜉,𝜕𝑟₁=𝜕𝜉𝜕𝑟₂𝜕𝜉=−1. (27) Hence substituting from (27) into (6), we get 𝜕𝑈𝑞𝜕𝜉=𝜉+¹1−𝜇𝑟²₁+𝑞₂𝜇𝑟²₂+1𝑐²𝜇−𝜇²𝑞−3𝜉−¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝑟²₁+𝑞₂𝜇𝑟²₂+12𝜉³+32𝑞₁1−𝜇𝑟²₁+𝑞₂𝜇𝑟²₂𝜉²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂𝜉−12𝑞₁𝑞₂𝜇×11−𝜇𝑟²₁−1𝑟²₁−1𝑟²₂1−1+3𝜇+7𝜉−7𝑟₁−1𝑟₂=0 (28) which can be rewritten as a function of 𝑟₁ and 𝑟₂ as 𝜕𝑈𝜕𝜉=1−𝜇−𝑟₂+𝑞₁1−𝜇𝑟²₁+𝑞₂𝜇𝑟²₂+1𝑐²𝜇−𝜇²−31−𝜇−𝑟₂−𝑞₁1−𝜇𝑟₁+𝑞₂𝜇𝑟₂×𝑞₁1−𝜇𝑟²₁+𝑞₂𝜇𝑟²₂+121−𝜇−𝑟₂³+32𝑞₁1−𝜇𝑟²₁+𝑞₂𝜇𝑟²₂×1−𝜇−𝑟₂²𝑞+3¹1−𝜇𝑟₁+𝑞₂𝜇𝑟₂1−𝜇−𝑟₂−12𝑞₁𝑞₂𝜇×11−𝜇𝑟²₁−1𝑟²₁−1𝑟²₂1−1+3𝜇+7𝜉−7𝑟₁−1𝑟₂=0. (29) Then it may be reasonable in our case to assume that the position of the equilibrium point 𝐿₃ is the same as given by classical restricted three-body problem but perturbed due to the inclusion of the relativistic correction by quantities 𝜀₃≡𝒪(1/𝑐²): 𝑟₁=𝑎₃+𝜀₃,𝑟₂=𝑏₃+𝜀₃,𝑎₃=𝑏₃−1, (30) where 𝑎₃ and 𝑏₃ are the unperturbed values of 𝑟₁ and 𝑟₂, respectively, and 𝑏₃ is given after some successive approximation by the relation 𝑏₃7=2−4𝑞₂𝑞₁𝜇3+21−𝑞₁+𝑞₂𝜇²4𝑞₂𝑞₂𝑞₁²𝜇3²+147−𝑞₁+𝑞₂16𝑞₂𝑞₂𝑞₁³𝜇3³+1029−𝑞₁+𝑞₂64𝑞₂𝑞₂𝑞₁⁴𝜇3⁴−7203𝑞₁−𝑞₂256𝑞₂𝑞₂𝑞₁⁵𝜇3⁵𝜇+𝒪3⁶. (31) Substituting from (30) into (29) and retaining the terms up to the first order in the small quantities 𝜀₃, we can get 𝜕𝑈𝜕𝜉=1−𝜇−𝑏₃−𝜀₃+𝑞₁1−𝜇1−𝑏₃²×1+2𝜀₃1−𝑏₃+𝑞₂𝜇𝑏²₃1−2𝜀₃𝑏₃+1𝑐²×𝜇−𝜇²−31−𝜇−𝑏₃−𝑞₁1−𝜇1−𝑏₃−𝑞₂𝜇𝑏₃𝑞₁1−𝜇1−𝑏₃²+𝑞₂𝜇𝑏²₃+121−𝜇−𝑏₃³+32𝑞₁1−𝜇1−𝑏₃²+𝑞₂𝜇𝑏²₃×1−𝜇−𝑏₃²𝑞−3¹1−𝜇1−𝑏₃−𝑞₂𝜇𝑏₃×1−𝜇−𝑏₃−12𝑞₁𝑞₂𝜇×11−𝜇1−𝑏₃²−11−𝑏₃²−1𝑏²₃×6−7𝑏₃1−4𝜇−71−𝑏₃−1𝑏₃=0. (32) Equation (32) can be solved for 𝜀₃; then substituting back into (30) and (27) yields, after some lengthy algebraic manipulations, the location of 𝐿₃ as 𝜉_0,𝐿3=1−𝜇−⁵_𝑘=0ℳ^𝐿3_𝑘+1𝑐²𝒩^𝐿3_𝑘𝜇3^𝑘𝜇+𝒪3⁶, (33) where the non-vanishing coefficients ℳ^𝐿3_𝑘 and 𝒩^𝐿3_𝑘 are collected in Appendix C.

Remark 2. It is worth noting that there is no contribution in other two coordinates; that is, 𝜂_0,𝐿𝑖=𝜁_0,𝐿𝑖=0,𝑖=1,2,3. This can be interpreted due to the fact that the concerned relativistic field is conservative.

5. Numerical Representations and Analyses

In this section we design a Mathematica program to draw the locations of collinear points 𝐿_𝑖,   𝑖=1,2,3, versus the whole range of the mass ratio 𝜇 taking into account the photogravitational effects and/or the relativistic corrections. In the case of one body radiates we considering the following values of the photogravitational parameter: 𝑞_𝑖=1,0.8,0.6,0.4,0.2, 𝑖=1,2, and they are represented on figures as black (no radiation exist), green, blue, orange, and red, respectively. But in the case two primaries radiating we adopted the following values, 𝑞_𝑖=1,0.6,0.2, 𝑖=1,2, and they are black (no radiation exist), blue, and red, respectively, this is actually to avoid the high density of the curves represented on these figures.

5.1. Analysis of 𝐿₁ Location

The black curve represents the case when the radiation pressure is totally neglected. In Figures 5(a) and 5(b), it seems that when 𝜇=0.5 the location of 𝐿₁ lies exactly in the barycenter, and when 𝜇→0 the location of 𝐿₁ is shifted towards the center of the less massive primary and then the problem reduced to the classical two-body problem. The same case also exists in Figure 5(c) when there is no radiation from the two primaries, but when one or two primaries radiate, the situation is changed.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5 

(a) The location of 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞1 and constant 𝑞2 without relativistic corrections. (b) The location of 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞2 and constant 𝑞1 without relativistic corrections. (c) The location of 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞1 and 𝑞2 without relativistic corrections. (d) The location of 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞1 and constant 𝑞2 with relativistic corrections. (e) The location of 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞2 and constant 𝑞1 with relativistic corrections. (f) The location of point 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞1 and 𝑞2 with relativistic corrections.

In Figure 5(a), we plot the location of 𝐿₁ under the effect of changing 𝑞₁ (the radiation coefficient of the massive primary 1−𝜇) with different mass ratios. The green, blue, orange, and red curves represent the effect of successive increase in radiation of the massive primary, respectively. As is cleared from these curves illustrate, the greater the effect of the radiation (mass loss) the greater the negative displacement of the location of 𝐿₁ away from less massive primary, that is, towards the massive primary. This effect is of nonlinear nature. When the mass loss is significantly large especially in the case of binary systems of comparable masses (near 𝜇=0.4 and greater), the location of 𝐿₁ crosses the barycenter towards the massive primary, that is, along the negative 𝜉-axis.

In Figure 5(b) we plot the location of 𝐿₁ under the effect of changing 𝑞₂ (the radiation coefficient of the less massive primary 𝜇) with different mass ratios; the greater the effect of the radiation (mass loss) the greater the positive displacement of the location of 𝐿₁ towards less massive primary. When the mass loss is significantly large especially in the case of binary systems of comparable masses (near 𝜇=0.4 and greater), the location of 𝐿₁ never crosses the barycenter towards the less massive primary. This may be interpreted as follows: as the emitter in this case is the less massive body, within the domain of classical stability 𝜇∈(0,0.038) the location of 𝐿₁ is not affected so much in contrast to the greater mass ratios.

In Figure 5(c), we plot the location of 𝐿₁ under the effect of changing 𝑞₁ and 𝑞₂ (the radiation coefficient of the two primaries). The situation is a combination of the two effects depicted in Figures 5(a) and 5(b). Since the point 𝐿₁ is closer to less massive primary, the effect of the radiation of it is dominant. With the observation that when the mass loss is significantly large especially in the case of binary systems of comparable masses (near 𝜇=0.4 and greater), the location of 𝐿₁ crosses the barycenter towards the massive primary.

Figures 5(d)–5(f) represent the location of Lagrangian point 𝐿₁ under the effect of changing 𝑞₁ and/or 𝑞₂ with relativistic corrections. In Figure 5(d) we plot the location of 𝐿₁ under the effect of changing 𝑞₁ (the radiation coefficient of the less massive primary 𝜇) with relativistic corrections versus different mass ratios. The greater the effect of the radiation (mass loss), the greater the positive displacement of the location of 𝐿₁ towards less massive primary in the domain 𝜇∈(0,0.038). The most observable thing is that 𝐿₁ crosses to the right of the less massive primary especially in the very small mass ratios. Beyond this domain, that is, when 𝜇∈(0.038,0.5), the situation converses, but the nonlinearity becomes more visible. In Figure 5(e), the dynamics is similar to that in Figure 5(b) but with clear different sizes of perturbations. In Figure 5(f), we have combined effects of those depicted in Figures 5(d) and 5(e).

5.2. Analysis of 𝐿₂ Location

The black curve represents the case when the radiation pressure is totally neglected. In Figures 6(a) and 6(b), it seems that when 𝜇=0.5 the location of 𝐿₂ lies exactly at 1.2, and when 𝜇→0 the location of 𝐿₂ is shifted to the left, that is, towards the center of the less massive primary and lies exactly at a normalized distance equal to 1 on the right of the barycenter. Then the problem reduced to the classical two body problem. The same case also exists in Figure 6(c) when there is no radiation from the two primaries, but when one or two primaries radiate, the situation is changed.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6 

(a) The location of 𝐿2 against mass ratio 𝜇 under the effect of changing 𝑞1 and constant 𝑞2 without relativistic corrections. (b) The location of 𝐿2 against mass ratio 𝜇 under the effect of changing 𝑞2 and constant 𝑞1 without relativistic corrections. (c) The location of 𝐿2 against mass ratio 𝜇 under the effect of changing 𝑞1and𝑞2 without relativistic corrections. (d) The location of 𝐿2 against mass ratio 𝜇 under the effect of changing 𝑞1 and constant 𝑞2 with relativistic corrections. (e) The location of 𝐿2 against mass ratio 𝜇 under the effect of changing 𝑞2 and constant 𝑞1 with relativistic corrections. (f) The location of 𝐿2 against mass ratio 𝜇 under the effect of changing 𝑞1 and 𝑞2 with relativistic corrections.

In Figure 6(a), we plot the location of 𝐿₂ under the effect of changing 𝑞₁ (the radiation coefficient of the massive primary 1−𝜇) versus different mass ratios. The green, blue, orange, and red curves represent the effect of successive increase in radiation of the massive primary, respectively. As these curves illustrate, the greater the effect of the radiation (mass loss), the greater the negative nonlinear displacement of the location of 𝐿₂. As is observed at high radiation and greater mass ratios specifically at 𝜇>0.038𝐿₂ crosses to the left of less massive primary. When 𝜇∈(0,0.2), the locations of 𝐿₂ have positive displacements, while this effect is reversed when 𝜇∈(0.2,0.5). In Figure 6(b) the situation is similar to that in Figure 6(b) but with different size of perturbations.

In Figure 6(c), we plot the location of 𝐿₂ under the effect of changing 𝑞₁ and 𝑞₂ (the radiation coefficient of the two primaries). The situation is a combination of the two effects depicted in Figures 6(a) and 6(b). Since the point 𝐿₂ is closer to less massive primary, the effect of the radiation of it is dominant.

Figures 6(d)–6(f) represent the location of Lagrangian point 𝐿₂ under the effect of changing 𝑞₁ and/or 𝑞₂ with relativistic corrections. In Figure 6(d) we plot the location of 𝐿₂ under the effect of changing 𝑞₁ (the radiation coefficient of the less massive primary 𝜇) with relativistic corrections versus different mass ratios. The greater the effect of the radiation (mass loss), the greater the positive displacement of the location of 𝐿₂ in the domain 𝜇∈(0,0.038). Beyond this domain, that is, when 𝜇∈(0.038,0.5), the situation converses, but the nonlinearity becomes more visible. In Figure 6(e), the dynamics is similar to that in Figure 6(b) but it is clear with different size of perturbations. In Figure 6(f), we have combined effects of those depicted in Figures 6(d), and 6(e).

5.3. Analysis of 𝐿₃ Location

The black curve represents the case when the radiation pressure is totally neglected. In Figure 7(a) and in Figure 7(b), it seems that when 𝜇=0.5 the location of 𝐿₃ lies exactly at a normalized distance equal to 1.2 to the left of the barycenter, and when 𝜇→0 the location of 𝐿₃ lies exactly at a normalized distance equal to 1 to the left of the barycenter, and then the problem reduced to the classical two body problem. The same case also exists in Figure 7(c) when there is no radiation from the two primaries, but when one or two primaries radiate, the situation is changed.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7 

(a) The location of 𝐿3 against mass ratio 𝜇 under the effect of changing 𝑞1 and constant 𝑞2 without relativistic corrections. (b) The location of 𝐿3 against mass ratio 𝜇 under the effect of changing 𝑞2 and constant 𝑞1 without relativistic corrections. (c) The location of 𝐿3 against mass ratio 𝜇 under the effect of changing 𝑞1 and 𝑞2 without relativistic corrections. (d) The location of 𝐿1 against mass ratio 𝜇 under the effect of changing 𝑞1 and constant 𝑞2 with relativistic corrections. (e) The location of 𝐿3 against mass ratio 𝜇 under the effect of changing 𝑞2 and constant 𝑞1 with relativistic corrections. (f) The location of 𝐿3 against mass ratio 𝜇 under the effect of changing 𝑞1 and 𝑞2 with relativistic corrections.

In Figure 7(a), we plot the location of 𝐿₃ under the effect of changing 𝑞₁ (the radiation coefficient of the massive primary 1−𝜇) against different mass ratios. The green, blue, orange, and red curves represent the effect of successive increase in radiation of the massive primary, respectively. As these curves illustrate, the greater the effect of the radiation (mass loss) the greater the positive displacement of the location of 𝐿₃, that is, towards the massive primary.

In the case of sever radiation (the red curve) and mass ratios 𝜇>0.15 the order of the primaries RTBP may be reversed; for example, when 𝑞₁>0.6 the location of 𝐿₃ is weakly nonlinearly decreased. When 𝑞₁≥0.8 and 0.15<𝜇<0.35 the location of 𝐿₃ is exponentially decreased while when 0.35<𝜇<0.5 the location of 𝐿₃ is exponentially increased.

In Figure 7(b) we plot the location of 𝐿₃ under the effect of changing 𝑞₂ (the radiation coefficient of the less massive primary 𝜇) against different mass ratios. The greater the effect of the radiation (mass loss), the greater the positive displacement of the location of 𝐿₃, linearly during 𝑞<0.4 and nonlinearly during 𝑞>0.4. This effect becomes very clear for significantly large mass ratios. When comparing the two most left and the two most right parts on Figure 7(a) and Figure 7(b), they seemed in contrast to each other when 𝑞<0.4 due to the fact that the location of 𝐿₃ is closer to the massive primary than the less massive primary.

The analysis of Figure 7(d) is that the greater the effect of the radiation (mass loss), the greater the negative displacement of the location of 𝐿₃; that is to say, for 𝑞<0.4 the analysis of Figure 7(d) is in weak contrast to the analysis of Figure 7(a), while for 𝑞>0.4 and significantly large mass ratios we have again positive displacements.

The analyses of Figure 7(e) are in strong contrast to that of Figure 7(b) but with different sizes of perturbations due to the inclusion of the relativistic effects in addition to the photogravitational effects.

Figures 7(c)–7(f) represent the location of Lagrangian point 𝐿₃ under the effect of changing 𝑞₁ and 𝑞₂ without/with relativistic corrections, respectively. They represent the combined effects of those depicted in Figures 7(a) and 7(b) and in Figures 7(d) and 7(e), respectively.

6. Conclusion

In a very important two works, Douskos and Perdios [7] and Ragos et al. [8] determined the stability of the equilibrium points and approximated the positions of the collinear points by series by expansions and determined the stability of the triangular as well as collinear points without taking into account the photo-gravitational effects of the two primaries. In our here presented work, we assumed the primaries to be radiant sources. The corrected locations of the collinear points are computed. Series forms of these location are obtained as new analytical results. In order to introduce a semianalytical view, A Mathematica program is constructed so as to draw the locations of collinear points versus the whole range of the mass ratio 𝜇 taking into account the photo-gravitational effects and/or the relativistic corrections. We analyzed all the obtained figures.

In a forthcoming works we aim to complete this work by performing the same computation to the triangular equilibrium points. Also we planned to study the stability of all equilibrium points; this would actually enhance the scientific significance of the current study.

Appendices

A.

The non-vanishing coefficients ℳ^𝐿1_𝑘 and 𝒩^𝐿1_𝑘, of (19) are as follows: ℳ^𝐿1₀=1−𝑞₁1+8𝑞₁,ℳ^𝐿1₁=3𝑞₁1+8𝑞₁²−1+28𝑞₁𝑞₂𝑞₁^1/3,ℳ^𝐿1₂=𝑞₁1+8𝑞₁³−7+417−76𝑞₁𝑞₁𝑞₂𝑞₁^2/3,ℳ^𝐿1₃=191+8𝑞₁⁴×27−1+𝑞₁1+8𝑞₁²−35+2𝑞₁−777+8𝑞₁213+292𝑞₁𝑞₂,ℳ^𝐿1₄=291+8𝑞₁⁵×27𝑞₁1+8𝑞₁²7+2𝑞₁−19+56𝑞₁+−11+𝑞₁1013+8𝑞₁×−753+4𝑞₁−317+884𝑞₁×𝑞₂𝑞₂𝑞₁^1/3,𝒩^𝐿1₀=121+8𝑞₁−1+𝑞₁5+2𝑞₁,𝒩^𝐿1₁=321+8𝑞₁²×1+𝑞₁31+6𝑞₁−56𝑞²₁𝑞₂𝑞₁^1/3,𝒩^𝐿1₂=11+8𝑞₁³×1+𝑞₁41+𝑞₁−153+172𝑞₁−1+4𝑞₁×𝑞₂𝑞₁^2/3,𝒩^𝐿1₃1=−18𝑞₁1+8𝑞₁⁴×27𝑞₁1+8𝑞₁²−5+𝑞₁3+4𝑞₁1+4𝑞₁+215+𝑞₁367+𝑞₁×11041+2𝑞₁16125+8𝑞₁×677+284𝑞₁𝑞₂,𝒩^𝐿1₄=118𝑞₁1+8𝑞₁⁵𝑞₂𝑞₁^1/3×27𝑞₁1+8𝑞₁²×5+2𝑞₁−10+𝑞₁231+8𝑞₁×−29+104𝑞₁+24+𝑞₁222+𝑞₁−4381+2𝑞₁×25441+8𝑞₁20607+4𝑞₁×5671+1740𝑞₁𝑞₂×𝑞₂𝑞₁^1/3,𝒩^𝐿1₅1=−54𝑞₁1+8𝑞₁⁶×27𝑞₁1+8𝑞₁²9+2𝑞₁×319+4𝑞₁25+𝑞₁×2523+8𝑞₁−23+412𝑞₁+2−9+2𝑞₁−228+𝑞₁×7285+𝑞₁−184181+16𝑞₁×−62741+𝑞₁−34747+8𝑞₁×10231+8644𝑞₁𝑞₂𝑞₂𝑞₁^2/3. (A.1)

B.

The non-vanishing coefficients ℳ^𝐿2_𝑘 and 𝒩^𝐿2_𝑘, of (26) are as follows: ℳ^𝐿2₀=−1+𝑞₁1+8𝑞₁ℳ^𝐿2₁=3𝑞₁1+8𝑞₁²−1+28𝑞₁𝑞₂𝑞₁^1/3ℳ^𝐿2₂=𝑞₁1+8𝑞₁³×7+4𝑞₁−17+76𝑞₁𝑞₂𝑞₁^2/3,ℳ^𝐿2₃1=−91+8𝑞₁⁴×27−1+𝑞₁1+8𝑞₁²+35+2𝑞₁−777+8𝑞₁×213+292𝑞₁𝑞₂,ℳ^𝐿2₄=291+8𝑞₁⁵×27𝑞₁1+8𝑞₁²7+2𝑞₁−19+56𝑞₁+11+𝑞₁−1013+8𝑞₁×753+4317−884𝑞₁×𝑞₁𝑞₂𝑞₂𝑞₁^1/3,ℳ^𝐿2₅=191+8𝑞₁⁶×9𝑞₁1+8𝑞₁²×−55+16𝑞₁24+𝑞₁×−279+304𝑞₁+21+𝑞₁×1287+16𝑞₁−778+𝑞₁×−397+8𝑞₁1131+236𝑞₁×𝑞₂𝑞₂𝑞₁^2/3,𝒩^𝐿2₀=12+16𝑞₁7+11𝑞₁1+2𝑞₁,𝒩^𝐿2₁=321+8𝑞₁²×1+𝑞₁25+2𝑞₁×7+12𝑞₁−3+16𝑞₁×𝑞₂𝑞₁^1/3,𝒩^𝐿2₂=11+8𝑞₁³×2+𝑞₁39+𝑞₁×847−48𝑞₁−51+𝑞₁×−89+64𝑞₁𝑞₂𝑞₁^2/3,𝒩^𝐿2₃=118𝑞₁1+8𝑞₁⁴×−27𝑞₁1+8𝑞₁²11+𝑞₁×67+4𝑞₁59+44𝑞₁+23+2𝑞₁123+𝑞₁×−957+𝑞₁15203+16𝑞₁×4475+9926𝑞₁+9216𝑞²₁𝑞₂,𝒩^𝐿2₄1=−18𝑞₁1+8𝑞₁⁵𝑞₂𝑞₁^1/3×27𝑞₁1+8𝑞₁²−5+2𝑞₁×−14+𝑞₁−699+8𝑞₁×−149+16𝑞₁23+84𝑞₁+27+𝑞₁343+4𝑞₁−382+𝑞₁×17073+8𝑞₁8977+8𝑞₁×2332+𝑞₁6521+9216𝑞₁𝑞₂,𝒩^𝐿2₅=154𝑞₁1+8𝑞₁⁶×27𝑞₁1+8𝑞₁²×3+2𝑞₁179+8𝑞₁×−278+𝑞₁−803+12𝑞₁×−949+12𝑞₁−123+128𝑞₁+2−9+2𝑞₁−219+𝑞₁×−15493+𝑞₁65081+16𝑞₁×2888+𝑞₁−39569+16𝑞₁×17815+2𝑞₁×38069+54528𝑞₁×𝑞₂𝑞₂𝑞₁^2/3. (B.1)

C.

The non-vanishing coefficients ℳ^𝐿3_𝑘 and 𝒩^𝐿3_𝑘 of (33) are as follows: ℳ^𝐿3₀=135+𝑞₁,ℳ^𝐿3₁=118−19−5𝑞₁+3𝑞₂,ℳ^𝐿3₂=796𝑞²₂−2𝑞²₁19+5𝑞₁+𝑞₁𝑞₂73+23𝑞₁−47+5𝑞₁𝑞²₂𝑞₂𝑞₁²,ℳ^𝐿3₃=72304𝑞³₂×−84𝑞³₁19+5𝑞₁+336𝑞²₁19+5𝑞₁𝑞₂−768𝑞₁11+4𝑞₁𝑞²₂+3248+1741𝑞₁𝑞³₂×𝑞₂𝑞₁³,ℳ^𝐿3₄=79216𝑞⁴₂×−588𝑞⁴₁19+5𝑞₁+414𝑞³₁19+5𝑞₁𝑞₂+9𝑞²₁5419+2173𝑞₁𝑞²₂−𝑞₁84371+46834𝑞₁𝑞³₂+56625+512𝑞₁𝑞⁴₂×𝑞₂𝑞₁⁴,ℳ^𝐿3₅=49147465𝑞⁵₂×−2352𝑞⁵₁19+5𝑞₁−6096𝑞⁴₁19+5𝑞₁𝑞₂+12𝑞³₁59527+22817𝑞₁𝑞²₂−12𝑞²₁96175+46304𝑞₁𝑞³₂+𝑞₁795713+420985𝑞₁𝑞⁴₂−214592+93715𝑞₁𝑞⁵₂×𝑞₂𝑞₁⁴,𝒩^𝐿3₀=165+𝑞₁−3+2𝑞₁,𝒩^𝐿3₁=17247−34𝑞²₁+15𝑞₂−91𝑞₂𝑞₁+𝑞₁15+8𝑞₂,𝒩^𝐿3₂1=−1728𝑞²₁×−2𝑞²₁−103+𝑞₁×537+50𝑞₁+𝑞₁×1855+𝑞₁579+4882𝑞₁𝑞₂+6−413+𝑞₁21+82𝑞₁𝑞²₂,𝒩^𝐿3₃=1124416𝑞³₁×12𝑞³₁1007+25𝑞₁−57+14𝑞₁+72𝑞²₁2163+𝑞₁×−1338+893𝑞₁𝑞₂+84𝑞₁×−3040+𝑞₁×2679+2615𝑞₁𝑞²₂−7−15092+𝑞₁×18417+1594𝑞₁𝑞³₂,𝒩^𝐿3₄=11492992𝑞⁴₁×84𝑞⁴₁1007+25𝑞₁×−57+14𝑞₁+6𝑞³₁357511+𝑞₁×−27069+205210𝑞₁𝑞₂−63𝑞²₁108347−7701𝑞₁+6582𝑞²₁𝑞²₂−7𝑞₁×−865669+2𝑞₁×8751+69208𝑞₁𝑞³₂+28−53648+𝑞₁×−8295+5113𝑞₁𝑞⁴₂,𝒩^𝐿3₅=771663616𝑞⁵₁×336𝑞⁵₁1007+25𝑞₁×−57+14𝑞₁+48𝑞⁴₁×135337+𝑞₁×29127+110680𝑞₁𝑞₂−108𝑞³₁285685+3𝑞₁×21979+84386𝑞₁𝑞²₂+12𝑞²₁4088959+2𝑞₁×505899+2124101𝑞₁×𝑞³₂−3𝑞₁11569257+𝑞₁×2427243+9605126𝑞₁𝑞⁴₂++71461824122055−169714𝑞₁𝑞₁𝑞⁵₂. (C.1)

Acknowledgments

The authors are deeply indebted and thankful to the deanship of the scientific research and his helpful and distinct team of employees at Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia. This research work was supported by Grant no. 3007/1434. The authors would also like to record their indebtedness and thankfulness to the reviewers for their valuable and fruitful comments as well as for their powerful reading and suggestions.