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

Authors: S. E. Abd El-Bar; F. A. Abd El-Salam
Journal: Mathematical Problems in Engineering
 (2013)

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## 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 ratioμ taking into account the photo-gravitational effects and/or the relativistic corrections. Analyses of these figures are addressed.


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## Body

## 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 withm1 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 atL2), 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 inμ and 1/c2 under the considered perturbations.The rest of the paper is organized as follows. In Section2 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 Li,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]:
(1)ξ¨-2nη˙=∂U∂ξ-ddt(∂U∂ξ˙),η¨+2nξ˙=∂U∂η-ddt(∂U∂η˙)
with
(2)n=1+12c2(μ(1-μ)-3),r=(ξ2+η2),
where U is the potential-like function of the relativistic RTBP, which can be written as composed of two components Uc+phand Ur+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:
(3)U=Uc+ph+Ur+ph,
where Uc+phand Ur+ph are given by
(4)Uc+ph=r22+q1(1-μ)r1+q2μr2,Ur+ph=r22c2(μ(1-μ)-3)+18c2×((ξ+η.)2+(η-ξ.)2)2+32c2(q1(1-μ)r1+q2μr2)×((ξ+η.)2+(η-ξ.)2)-12c2(q1(1-μ)r1+q2μr2)2-q1q2μ(1-μ)2c2×[(q2μr13+q1(1-μ)r23)1r1+(1r1-1r2)(1-3μ-7ξ-8η.)+η2(q2μr13+q1(1-μ)r23)],r1=(ξ+μ)2+η2,r2=(ξ+μ-1)2+η2,
where qi=1-(Fp/Fg)i,i=1,2, is the mass reduction factor constant for a given particle. Fp, is the force due to radiation pressure, and Fg 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 qi.


## 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 (Uξ,Uη,Uζ) 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 theξ-η plane, that is, in the plane of motion.The libration points are obtained from equations of motion after settingξ¨=η¨=ξ˙=η˙=0.These points represent particular solutions of equations of motion
(5)∂U∂ξ=∂Uc+ph∂ξ+∂Ur+ph∂ξ=0,∂U∂η=∂Uc+ph∂η+∂Ur+ph∂η=0.
The explicit required partial derivatives are (remembering that U=Uc+ph+Ur+ph)
(6)∂U∂ξ=ξ-q1(1-μ)(ξ+μ)r13-q2μ(ξ+μ-1)r23+1c2×{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)(ξ+μ)r13+q2μ(ξ+μ-1)r23)+12(η2+ξ2)ξ-32×(q1(1-μ)(ξ+μ)r13+q2μ(ξ+μ-1)r23)×(η2+ξ2)+3(q1(1-μ)r1+q2μr2)ξ-12q1q2μ(1-μ)×[-1r13(ξ+μ)+((ξ+μ)r13-(ξ+μ-1)r23)×(-1+3μ+7ξ)-7(1r1-1r2)-3η2×(q2μ(ξ+μ)r15+q1(1-μ)(ξ+μ-1)r25)]}=0,(7)∂U∂η=η-q1(1-μ)r13η-q2μr23η+1c2{(μ-μ2-3)η+(q1(1-μ)r1+q2μr2)(q2μr13+q1(1-μ)r23)×(q1(1-μ)r13+q2μr23)η+12(ξ2+η2)η-32(ξ2+η2)(q1(1-μ)r13+q2μr23)η+3(q1(1-μ)r1+q2μr2)η+q1q2μ(1-μ)2η+12q1q2μ(1-μ)η×[(-1r13+1r23)(-1+3μ+7ξ)+3(q1(1-μ)r25+q2μr15)η2+1r13-2(q2μr13+q1(1-μ)r23)]}=0.


## 4. Location of Collinear Libration Points

Any point of the collinear points must, by definition, haveζ=η=0, and the solution of the classical RTBP satisfies
(8)σ1r1+σ2r2=1,r1=σ1(ξ+μ),r2=-σ2(μ+ξ-1),
where (see Figures 2, 3, and 4) we have
(9)L1:σ1=1,σ2=1,L2:σ1=1,σ2=-1,L3:σ1=-1,σ2=1.Figure 2
The Location ofL1.Figure 3
The Location ofL2.Figure 4
The Location ofL3.


### 4.1. Location ofL1

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
(10)∂U∂ξ=ξ-(q1(1-μ)r12-q2μr22)+1c2×{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12-q2μr22)-32(q1(1-μ)r12-q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ+12ξ3-12q1q2μ(1-μ)×[-1r12+(1r12+1r22)(-1+3μ+7ξ)(q1(1-μ)r1+q2μr2)-7(1r1-1r2)]}=0
which can be re-written as a function of r1,r2 as
(11)∂U∂ξ=1-μ-r2-(q1(1-μ)r12-q2μr22)+1c2{(q1(1-μ)r1+q2μr2)(q1(1-μ)r12-q2μr22)1r12+(μ-μ2-3)×(1-μ-r2)+12(1-μ-r2)3-32(q1(1-μ)r12-q2μr22)×(1-μ-r2)2+3(q1(1-μ)r1+q2μr2)×(1-μ-r2)-12q1q2μ(1-μ)×[-1r12+(1r12+1r22)×(-1+3μ+7(1-μ-r2))1r12-7(1r1-1r2)]}=0.
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 Uc+ph=q1(1-μ)(r12+2/r1)+q2μ(r22+2/r2) as follows:
(12)∂Uc+ph∂ξ=∂Uc+ph∂r1∂r∂ξ+∂Uc+ph∂r2∂r2∂ξ=0,⟹∂Uc+ph∂ξ=∂Uc+ph∂r1-∂Uc+ph∂r2=0⟹∂Uc+ph∂r1=∂Uc+ph∂r2⟹q2μq1(1-μ)=(3r2-3r22+r23)r22(1-r2)2(1-r23).
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
(13)α=r2(1+13r2+13r22+5381r23+101243r24+𝒪(r25)+⋯).
Let successive approximations of r2 be r2i:
(14)r20=α,α=r21(1+13α),r21=α(1-13α),α=r22[1+13α+29α2+⋯],⋮
Then successive approximation yields
(15)r24=α(1-α3-α29+227α3+281α4+𝒪(α)5).
Let us set the following notation:
(16)r1=a1+ε1,r2=b1-ε1,a1=1-b1,
where a1 and b1 are unperturbed relativistically positions of r1 and r2, respectively, and b1 is given after some successive approximation by the relation
(17)b1=r24=α(1-α3-α29+227α3+281α4+𝒪(α)5).
Substituting from (16) into (11) and retaining the terms up to the first order in the small quantities ε1, we can get
(18)∂U∂ξ=1-μ-b1+ε1-q1(1-μ)(1-b1)2(1-2ε11-b1)+q2μb12(1+2ε1b1)+1c2×{(μ-μ2-3)(1-μ-b1)1(1-b1)2+(q1(1-μ)(1-b1)+q2μb1)(q1(1-μ)(1-b1)2-q2μb12)+12(1-μ-b1)3-32×(q1(1-μ)(1-b1)2-q2μb12)×(1-μ-b1)2+3(q1(1-μ)(1-b1)+q2μb1)×(1-μ-b1)-12q1q2μ(1-μ)×[-1(1-b1)2-7(1(1-b1)-1b1)+(1(1-b1)2+1b12)1(1-b1)2×(6-7b1-4μ)]}=0.
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
(19)ξ0,L1=1-μ-∑k=05[ℳkL1+(1c2)𝒩kL1](μ3)k/3+𝒪(μ3)2,
where the nonvanishing coefficients ℳkL1 and 𝒩kL1, are collected in Appendix A.


### 4.2. Location ofL2

The collinear pointL2 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
(20)r1-r2=1,r1=ξ+μ,r2=ξ+μ-1,∂r1∂ξ=∂r2∂ξ=1.
Substituting from (20) into (6), we get
(21)∂U∂ξ=ξ-(q1(1-μ)r12-q2μr22)+1c2×{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12-q2μr22)-32(q1(1-μ)r12-q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ+12ξ3-12q1q2μ(1-μ)×[-1r12+(1r12-1r22)×(-1+3μ+7ξ)-7(1r1-1r2)(1r12-1r22)]}=0
which can be rewritten as a function of r1 and r2 as
(22)∂U∂ξ=1-μ+r2-(q1(1-μ)r12-q2μr22)+1c2{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12-q2μr22)-32(q1(1-μ)r12-q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ+12ξ3-12q1q2μ(1-μ)×[-1r12+(1r12-1r22)(q1(1-μ)r1+q2μr2)×(-1+3μ+7ξ)-7(1r1-1r2)]}=0.
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):
(23)r1=a2+ε2,r2=b2+ε2,a2=1+b2
with a2 and b2 being the unperturbed positions  of r1 and r2, respectively, where b2 is given after some successive approximation as
(24)b2=α(1+α3-α29-227α3+281α4+𝒪(α)5),α=(q2μ3q1(1-μ))1/3.
Substituting from (23) into (22) and retaining the terms up to the first order in the small quantities ε2, we can get
(25)∂U∂ξ=1-μ+b2+ε2-q1(1-μ)(1+b2)2×(1-2ε21+b2)-q2μb22(1-2ε2b2)+1c2{(μ-μ2-3)(1-μ+b2)[1(1+b2)2+(q1(1-μ)(1+b2)+q2μb2)×(q1(1-μ)(1+b2)2+q2μb22)+12(1-μ+b2)3-32(q1(1-μ)(1+b2)2+q2μb22)×(1-μ+b2)2+3×(q1(1-μ)(1+b2)+q2μb2)×(1-μ+b2)-12q1q2μ(1-μ)×[-1(1+b2)2-7(1(1+b2)-1b2)+(1(1+b2)2-1b22)(6+7b2-4μ)]}=0.
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
(26)ξ0,L2=1-μ+∑k=05[ℳkL2+(1c2)𝒩kL2](μ3)k/3+𝒪(μ3)2,
where the non-vanishing coefficients ℳkL2 and 𝒩kL2, are collected in Appendix B.


### 4.3. Location ofL3

TheL3 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
(27)r2-r1=1,r1=-(ξ+μ),r2=1-μ-ξ,∂r1∂ξ=∂r2∂ξ=-1.
Hence substituting from (27) into (6), we get
(28)∂U∂ξ=ξ+(q1(1-μ)r12+q2μr22)+1c2{(μ-μ2-3)ξ-(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12+q2μr22)+12ξ3+32(q1(1-μ)r12+q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ-12q1q2μ(1-μ)×[1r12-(1r12-1r22)(-1+3μ+7ξ)-7(1r1-1r2)(q1(1-μ)r1+q2μr2)]}=0
which can be rewritten as a function of r1 and r2 as
(29)∂U∂ξ=1-μ-r2+(q1(1-μ)r12+q2μr22)+1c2{(μ-μ2-3)(1-μ-r2)(1r1-1r2)-(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12+q2μr22)+12(1-μ-r2)3+32(q1(1-μ)r12+q2μr22)×(1-μ-r2)2+3(q1(1-μ)r1+q2μr2)(1-μ-r2)-12q1q2μ(1-μ)×[1r12-(1r12-1r22)(-1+3μ+7ξ)-7(1r1-1r2)(1r12-1r22)]}=0.
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):
(30)r1=a3+ε3,r2=b3+ε3,a3=b3-1,
where a3 and b3 are the unperturbed values of r1 and r2, respectively, and b3 is given after some successive approximation by the relation
(31)b3=2-74(q2q1)(μ3)+21(-q1+q2)μ24q2(q2q1)2(μ3)2+147(-q1+q2)16q2(q2q1)3(μ3)3+1029(-q1+q2)64q2(q2q1)4(μ3)4-7203(q1-q2)256q2(q2q1)5(μ3)5+𝒪(μ3)6.
Substituting from (30) into (29) and retaining the terms up to the first order in the small quantities ε3, we can get
(32)∂U∂ξ=1-μ-b3-ε3+q1(1-μ)(1-b3)2×(1+2ε31-b3)+q2μb32(1-2ε3b3)+1c2×{(μ-μ2-3)(1-μ-b3)1(1-b3)2-(q1(1-μ)(1-b3)-q2μb3)(q1(1-μ)(1-b3)2+q2μb32)+12(1-μ-b3)3+32(q1(1-μ)(1-b3)2+q2μb32)×(1-μ-b3)2-3(q1(1-μ)(1-b3)-q2μb3)×(1-μ-b3)-12q1q2μ(1-μ)×[1(1-b3)2-(1(1-b3)2-1b32)1(1-b3)2×(6-7b3-4μ)-7(1(1-b3)-1b3)]}=0.
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
(33)ξ0,L3=1-μ-∑k=05[ℳkL3+(1c2)𝒩kL3](μ3)k+𝒪(μ3)6,
where the non-vanishing coefficients ℳkL3  and  𝒩kL3 are collected in Appendix C.Remark 2.
It is worth noting that there is no contribution in other two coordinates; that is,η0,Li=ζ0,Li=0,i=1,2,3. This can be interpreted due to the fact that the concerned relativistic field is conservative.


## 4.1. Location ofL1

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
(10)∂U∂ξ=ξ-(q1(1-μ)r12-q2μr22)+1c2×{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12-q2μr22)-32(q1(1-μ)r12-q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ+12ξ3-12q1q2μ(1-μ)×[-1r12+(1r12+1r22)(-1+3μ+7ξ)(q1(1-μ)r1+q2μr2)-7(1r1-1r2)]}=0
which can be re-written as a function of r1,r2 as
(11)∂U∂ξ=1-μ-r2-(q1(1-μ)r12-q2μr22)+1c2{(q1(1-μ)r1+q2μr2)(q1(1-μ)r12-q2μr22)1r12+(μ-μ2-3)×(1-μ-r2)+12(1-μ-r2)3-32(q1(1-μ)r12-q2μr22)×(1-μ-r2)2+3(q1(1-μ)r1+q2μr2)×(1-μ-r2)-12q1q2μ(1-μ)×[-1r12+(1r12+1r22)×(-1+3μ+7(1-μ-r2))1r12-7(1r1-1r2)]}=0.
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 Uc+ph=q1(1-μ)(r12+2/r1)+q2μ(r22+2/r2) as follows:
(12)∂Uc+ph∂ξ=∂Uc+ph∂r1∂r∂ξ+∂Uc+ph∂r2∂r2∂ξ=0,⟹∂Uc+ph∂ξ=∂Uc+ph∂r1-∂Uc+ph∂r2=0⟹∂Uc+ph∂r1=∂Uc+ph∂r2⟹q2μq1(1-μ)=(3r2-3r22+r23)r22(1-r2)2(1-r23).
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
(13)α=r2(1+13r2+13r22+5381r23+101243r24+𝒪(r25)+⋯).
Let successive approximations of r2 be r2i:
(14)r20=α,α=r21(1+13α),r21=α(1-13α),α=r22[1+13α+29α2+⋯],⋮
Then successive approximation yields
(15)r24=α(1-α3-α29+227α3+281α4+𝒪(α)5).
Let us set the following notation:
(16)r1=a1+ε1,r2=b1-ε1,a1=1-b1,
where a1 and b1 are unperturbed relativistically positions of r1 and r2, respectively, and b1 is given after some successive approximation by the relation
(17)b1=r24=α(1-α3-α29+227α3+281α4+𝒪(α)5).
Substituting from (16) into (11) and retaining the terms up to the first order in the small quantities ε1, we can get
(18)∂U∂ξ=1-μ-b1+ε1-q1(1-μ)(1-b1)2(1-2ε11-b1)+q2μb12(1+2ε1b1)+1c2×{(μ-μ2-3)(1-μ-b1)1(1-b1)2+(q1(1-μ)(1-b1)+q2μb1)(q1(1-μ)(1-b1)2-q2μb12)+12(1-μ-b1)3-32×(q1(1-μ)(1-b1)2-q2μb12)×(1-μ-b1)2+3(q1(1-μ)(1-b1)+q2μb1)×(1-μ-b1)-12q1q2μ(1-μ)×[-1(1-b1)2-7(1(1-b1)-1b1)+(1(1-b1)2+1b12)1(1-b1)2×(6-7b1-4μ)]}=0.
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
(19)ξ0,L1=1-μ-∑k=05[ℳkL1+(1c2)𝒩kL1](μ3)k/3+𝒪(μ3)2,
where the nonvanishing coefficients ℳkL1 and 𝒩kL1, are collected in Appendix A.


## 4.2. Location ofL2

The collinear pointL2 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
(20)r1-r2=1,r1=ξ+μ,r2=ξ+μ-1,∂r1∂ξ=∂r2∂ξ=1.
Substituting from (20) into (6), we get
(21)∂U∂ξ=ξ-(q1(1-μ)r12-q2μr22)+1c2×{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12-q2μr22)-32(q1(1-μ)r12-q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ+12ξ3-12q1q2μ(1-μ)×[-1r12+(1r12-1r22)×(-1+3μ+7ξ)-7(1r1-1r2)(1r12-1r22)]}=0
which can be rewritten as a function of r1 and r2 as
(22)∂U∂ξ=1-μ+r2-(q1(1-μ)r12-q2μr22)+1c2{(μ-μ2-3)ξ+(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12-q2μr22)-32(q1(1-μ)r12-q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ+12ξ3-12q1q2μ(1-μ)×[-1r12+(1r12-1r22)(q1(1-μ)r1+q2μr2)×(-1+3μ+7ξ)-7(1r1-1r2)]}=0.
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):
(23)r1=a2+ε2,r2=b2+ε2,a2=1+b2
with a2 and b2 being the unperturbed positions  of r1 and r2, respectively, where b2 is given after some successive approximation as
(24)b2=α(1+α3-α29-227α3+281α4+𝒪(α)5),α=(q2μ3q1(1-μ))1/3.
Substituting from (23) into (22) and retaining the terms up to the first order in the small quantities ε2, we can get
(25)∂U∂ξ=1-μ+b2+ε2-q1(1-μ)(1+b2)2×(1-2ε21+b2)-q2μb22(1-2ε2b2)+1c2{(μ-μ2-3)(1-μ+b2)[1(1+b2)2+(q1(1-μ)(1+b2)+q2μb2)×(q1(1-μ)(1+b2)2+q2μb22)+12(1-μ+b2)3-32(q1(1-μ)(1+b2)2+q2μb22)×(1-μ+b2)2+3×(q1(1-μ)(1+b2)+q2μb2)×(1-μ+b2)-12q1q2μ(1-μ)×[-1(1+b2)2-7(1(1+b2)-1b2)+(1(1+b2)2-1b22)(6+7b2-4μ)]}=0.
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
(26)ξ0,L2=1-μ+∑k=05[ℳkL2+(1c2)𝒩kL2](μ3)k/3+𝒪(μ3)2,
where the non-vanishing coefficients ℳkL2 and 𝒩kL2, are collected in Appendix B.


## 4.3. Location ofL3

TheL3 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
(27)r2-r1=1,r1=-(ξ+μ),r2=1-μ-ξ,∂r1∂ξ=∂r2∂ξ=-1.
Hence substituting from (27) into (6), we get
(28)∂U∂ξ=ξ+(q1(1-μ)r12+q2μr22)+1c2{(μ-μ2-3)ξ-(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12+q2μr22)+12ξ3+32(q1(1-μ)r12+q2μr22)ξ2+3(q1(1-μ)r1+q2μr2)ξ-12q1q2μ(1-μ)×[1r12-(1r12-1r22)(-1+3μ+7ξ)-7(1r1-1r2)(q1(1-μ)r1+q2μr2)]}=0
which can be rewritten as a function of r1 and r2 as
(29)∂U∂ξ=1-μ-r2+(q1(1-μ)r12+q2μr22)+1c2{(μ-μ2-3)(1-μ-r2)(1r1-1r2)-(q1(1-μ)r1+q2μr2)×(q1(1-μ)r12+q2μr22)+12(1-μ-r2)3+32(q1(1-μ)r12+q2μr22)×(1-μ-r2)2+3(q1(1-μ)r1+q2μr2)(1-μ-r2)-12q1q2μ(1-μ)×[1r12-(1r12-1r22)(-1+3μ+7ξ)-7(1r1-1r2)(1r12-1r22)]}=0.
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):
(30)r1=a3+ε3,r2=b3+ε3,a3=b3-1,
where a3 and b3 are the unperturbed values of r1 and r2, respectively, and b3 is given after some successive approximation by the relation
(31)b3=2-74(q2q1)(μ3)+21(-q1+q2)μ24q2(q2q1)2(μ3)2+147(-q1+q2)16q2(q2q1)3(μ3)3+1029(-q1+q2)64q2(q2q1)4(μ3)4-7203(q1-q2)256q2(q2q1)5(μ3)5+𝒪(μ3)6.
Substituting from (30) into (29) and retaining the terms up to the first order in the small quantities ε3, we can get
(32)∂U∂ξ=1-μ-b3-ε3+q1(1-μ)(1-b3)2×(1+2ε31-b3)+q2μb32(1-2ε3b3)+1c2×{(μ-μ2-3)(1-μ-b3)1(1-b3)2-(q1(1-μ)(1-b3)-q2μb3)(q1(1-μ)(1-b3)2+q2μb32)+12(1-μ-b3)3+32(q1(1-μ)(1-b3)2+q2μb32)×(1-μ-b3)2-3(q1(1-μ)(1-b3)-q2μb3)×(1-μ-b3)-12q1q2μ(1-μ)×[1(1-b3)2-(1(1-b3)2-1b32)1(1-b3)2×(6-7b3-4μ)-7(1(1-b3)-1b3)]}=0.
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
(33)ξ0,L3=1-μ-∑k=05[ℳkL3+(1c2)𝒩kL3](μ3)k+𝒪(μ3)6,
where the non-vanishing coefficients ℳkL3  and  𝒩kL3 are collected in Appendix C.Remark 2.
It is worth noting that there is no contribution in other two coordinates; that is,η0,Li=ζ0,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 pointsLi,   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: qi=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, qi=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 ofL1 Location

The black curve represents the case when the radiation pressure is totally neglected. In Figures5(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) The location ofL1 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.
(a)
(b)
(c)
(d)
(e)
(f)In Figure5(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 Figure5(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 Figure5(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.Figures5(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 ofL2 Location

The black curve represents the case when the radiation pressure is totally neglected. In Figures6(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) The location ofL2 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.
(a)
(b)
(c)
(d)
(e)
(f)In Figure6(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 Figure6(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.Figures6(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 ofL3 Location

The black curve represents the case when the radiation pressure is totally neglected. In Figure7(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) The location ofL3 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.
(a)
(b)
(c)
(d)
(e)
(f)In Figure7(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 ratiosμ>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 Figure7(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 Figure7(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 Figure7(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.Figures7(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.


## 5.1. Analysis ofL1 Location

The black curve represents the case when the radiation pressure is totally neglected. In Figures5(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) The location ofL1 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.
(a)
(b)
(c)
(d)
(e)
(f)In Figure5(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 Figure5(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 Figure5(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.Figures5(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 ofL2 Location

The black curve represents the case when the radiation pressure is totally neglected. In Figures6(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) The location ofL2 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.
(a)
(b)
(c)
(d)
(e)
(f)In Figure6(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 Figure6(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.Figures6(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 ofL3 Location

The black curve represents the case when the radiation pressure is totally neglected. In Figure7(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) The location ofL3 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.
(a)
(b)
(c)
(d)
(e)
(f)In Figure7(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 ratiosμ>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 Figure7(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 Figure7(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 Figure7(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.Figures7(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.



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*Source: 794734-2013-09-22.xml*