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formerly Central European Journal of Physics

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Volume 16, Issue 1

Issues

Volume 13 (2015)

The collinear equilibrium points in the restricted three body problem with triaxial primaries

Sultan Z. Alamri / Sobhy E. Abd El-Bar
  • Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, KSA, Saudi Arabia
  • Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt
  • Other articles by this author:
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/ Aly R. Seadawy
  • Corresponding author
  • Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, Saudi Arabia
  • Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
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Published Online: 2018-09-06 | DOI: https://doi.org/10.1515/phys-2018-0069

Abstract

The perturbed restricted three body problem has been reviewed. The mass of the primaries are assumed as triaxial. The locations of the collinear points have been computed. Series forms of these locations are obtained as new analytical results. In order to introduce a semi-analytical view, a Mathematica program has been constructed to graph the locations of collinear points versus the whole range of the mass ratio μ taking into account the triaxiality. The resultant figures have been analyzed

Keywords: Restricted three body problem; Collinear equilibrium points; perturbation; Triaxial; oblateness

PACS: 95.10.Ce; 45.50.Pk; 98.10.+z

1 Introduction

The restricted three-body problem (RTBP) is defined as a system where an infinitesimal mass, m3, is attracted gravitationally by two finite arbitrary masses called primaries, m1 and m2, but their motion is not influenced. The term restricted comes from the fact that all masses are assumed to move in the same plane defined by the two revolving primaries which revolve around their center of mass in circular orbits. This definition is widely used in almost all classical books of celestial mechanics, e.g. Szebehely [1] and Murray and Drmott [2]. For a more generalized version of the problem many authors have amended the potential function with some relevant perturbing forces; e.g. considering oblate primaries instead of spherical masses. Or even more generalized as triaxial bodies, inclusion of the relativistic effects, assuming the primaries are emitters, and/or move in a resisting medium. Even if the RTBP is not integrable, a number of special solutions can be found in the rotating frame where the third body has zero velocity and zero acceleration. These solutions correspond to equilibrium 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, three of them are collinear points, namely L1, L2, L3 and the another two are triangular points, namely L4, L5. The position of the infinitesimal body is displaced a little from the equilibrium point due to the some perturbations. If the resultant motion of the infinitesimal mass is a rapid departure from the vicinity of the point, we can call such a position of equilibrium point an “unstable” one, if however the body merely oscillates about the equilibrium point, it is said to be a “stable position” (in the sense of Lyapunov), Abd El-Salam [3]. In general, the dynamics of a circular and/or elliptical three-body problem is widely applicable toastrophysics, for example stellar/solar system dynamics and Earth-Moon system.This problem consequently received more attention from astronomers and dynamical system scientists. In spite of it, the solutions of this problem has been developed over the past centuries.

The literature concerning RTBPis extensive and it is worth highlighting here some relevant and recent studies dealingwith RTBP, with and without considering different perturbations: Sharma [4], Tsirogiannis et al. [5], Kushvah and Ishwar [6], Vishnu Namboori et al. [7], Mital et al. [8], Kumar and Ishwar [9], Rahoma and Abd El-Salam [10] and references therein. Ammar [11] analyzed solar radiation pressure effect on the positions and stability of the libration points in elliptic RTBP. Singh [12] formulated the triangular librationpoints nonlinear stability under the Coriolis effect and centrifugal forces as small perturbations in addition to the effect of primaries oblateness and radiation pressures. Singh and Umar [13] investigated the effect of luminous and oblate spheroids primaries on the locations and stability of the collinear libration points. In another work, Singh and Umar [14] studied the effect of the big primary’s triaxial and spherical shape of the companion on the locations and stability of the collinear libration points. They found that the position of collinear libration points and their stability are affected with their considered perturbations in addition to the eccentricity and the semi-major axis of the primaries orbit.

Katour et al. [15], Singh and Bello [16, 17], Abd El-Salam and Abd El-Bar [18], Abd El-Bar et al. [19] and Bello and Singh [20] were concerned with the relativistic RTBP in addition to some different perturbations—the primaries oblateness andradiation from one of the primaries—upon the equilibrium points locations and stability. They noticed that the stability regions of the concerned equilibrium points are varied (expanding or shrinking) related to the critical mass value and depending upon the value of their considered perturbations.

The fundamental structure in nature and science such as RTBP, plasma containment in tokamaks and stellarators for energy generation, population ecology, chaoticbehavior in biological systems, neural networks and solitonicfibre optical communication devices can be expound by nonlinear partial differential equations [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37].

The aim of this study is the determination of the locations of the collinear equilibrium points, taking into consideration the fact that both primaries are triaxial. This paper will be organized as follows: Section 1 is an historical introduction, the equations of motion in an RTBP with triaxial primaries are formulated in Section 2, the computations related to the location of equilibrium points with the considered perturbations are introduced in section 3. Section 4 highlightssomenumerical simulations with a discussion and forthcoming works. The paper finishes with a conclusion in Section 5.

2 Equations of motion

We shall adopt the notation and terminology of Szebehely [1]. As a consequence, the distance between the primaries m1 and m2 does not change and is taken equal to one; the sum of the masses of the primaries is also taken as one. The unit of time is also chosen as to make the gravitational constant unity.

The equations of motion of the infinitesimal mass m3 in the RTBP in a synodic co-ordinate system(x, y)in dimensionless variables in which the primary coordinates on the x-axis (–μ, 0), (1 – μ, 0) are given by Brumberg (1972).

x¨2ny˙=UxddtUx˙,y¨+2nx˙=UyddtUy˙(1)

where U is the potential–like function of the RTBP, which can be written as composed of two components (compound), namely the potential of the classical RTBP potential U are given by:

U=n22(1μ)r12+μr22+(1μ)r1+μr2+(1μ)(2σ1σ2)2r133(1μ)(σ1σ2)2r15y2+μ(2γ1γ2)2r233μ(γ1γ2)2r25y2

with

r=x2+y2,r1=(x+μ)2+y2r2=(x+μ1)2+y2,μ=m2m1+m212,

m1, m2(m1m2) being the masses of the primaries,

σ1=a12c125R2,σ2=b12c125R2,σ1,σ21,γ1=a22c225R2,γ2=b22c225R2,γ1,γ21,

ai, bi, ci, i = 1, 2 are the semi-axes of the triaxial of two primaries respectively and R the distance between the primaries.

The mean motion n of the primaries is given by

n2=1+322σ1σ2+322γ1γ2

Remark

The oblateness case of the primaries can be deduced directly from applying the condition ai = bi, i = 1, 2 or alternatively σ1 = σ2.

The libration points are obtained from equations of motion after setting = ÿ = = = 0. These points represent particular solutions of equations of motion

Ux=Uy=0,(2)

The explicit formulas are

Ux=n2x+(x+μ)(1μ)r133(1μ)(2σ1σ2)2r15+15(1μ)(σ1σ2)2r17y2+(x+μ1)μr233μ(2γ1γ2)2r25+15μ(γ1γ2)2r27y2(3)

and

Uy=n2y(1μ)r13+3(1μ)(2σ1σ2)2r15+3(1μ)(σ1σ2)2r15y(4)

3 Location of collinear libration points

Any point of the collinear points must, by definition, have z = y = 0, and the solution of the classical RTBP satisfies the conditions of Abd El-Bar and Abd El-Salam [38]

B1r1+B2r2=1,r1=B1x+μ,r2=B2μ+x1(5)

where, we have (see Figures 1, 2, 3):

The location of L1 and its corresponding parameters
Figure 1

The location of L1 and its corresponding parameters

The location of L2 and its corresponding parameters
Figure 2

The location of L2 and its corresponding parameters

The location of L3 and its corresponding parameters
Figure 3

The location of L3 and its corresponding parameters

L1:B1=1,B2=1,L2:B1=1,B2=1,L3:B1=1,B2=1

4 Location of L1

The location geometry of L1 can be visualized as given by Figure 1.

Substituting from (5) with the corresponding values of 𝓑1 = 𝓑2 = 1, we get r1 + r2 = 1, r1 = x + μ, r2 = 1 – μx and noting that r1x=r2x=1 , equation (3) becomes:

Ux=n2(1μr2)(1μ)r13+3(1μ)(2σ1σ2)2r1515(1μ)(σ1σ2)2r17y2+μr23+3μ2γ1γ22r2515μγ1γ22r27y2(6)

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 triaxial primaries

r1=a1+ε1,r2=b1ε1,a1=1b1(7)

from which we have

r1=1b11+ε11b1r2=b11ε1b1(8)

where a1 and b1 are unperturbed positions of r1 and r2 respectively, and b1 is given after some successive approximation by

b1=α1α3α29+227α3+281α4,α=μ3(1μ)3(9)

Substituting from equations (7) into equation (6)

Ux=n21μb1+ε11μ(1b1)212ε11b13(1μ)(2σ1σ2)2(1b1)4(14ε11b1)+15(1μ)(σ1σ2)2(1b1)616ε11b1y2+μb121+2ε1b1+3μ(2γ1γ2)2b141+4ε1b115μ(γ1γ2)2b161+6ε1b1y2

Since y = 0, then the above equation may be written as

Ux=n2(1μb1+ε1)1μ(1b1)212ε11b13(1μ)(2σ1σ2)2(1b1)414ε11b1+15(1μ)(σ1σ2)2(1b1)616ε11b1y2+μb121+2ε1b1+3μ(2γ1γ2)2b141+4ε1b1=0

which can be solved for ε1 yielding

ε1=n2+21μ1b132μb136μ2γ1γ2b15+61μ2σ1σ21b151n2+n2μ+1μ1b12μb12+n2b13μ2γ1γ22b14+31μ2σ1σ221b14(10)

Setting

d1=n2+21μ1b132μb136μ2γ1γ2b15+61μ2σ1σ21b151;e1=1b12,f1=1b14,g1=1(1b1)2,h1=1(1b1)4

Using the above relations, equation (10) can be written in the form

ε1=d1(n2(1+μ+b1)+1μg1μe13μ2γ1γ22f1+31μ2σ1σ22h1)(11)

where b1, d1, e1, f1, g1 and h1 are function of μ and they are given by

b1=μ31/313μ32/319μ3+2927μ34/35281μ35/313μ32+6227μ37/312581μ38/3μ33+Oμ310/3d1=i=09Ni1μ3i/3e1=59+233μ13+3μ23169μ3135081μ32310243μ3608729μ343148243μ3539476561μ322204819683μ3732104419683μ38387034177147μ33+f1=7327+331/3μ4/3+4μ+14331/3μ2/368932/3μ1/3500μ1/324331/3+4μ2/3332/3+1324μ6561434μ4/31968331/3+35180μ5/35904932/3+13294μ259049+339800μ7/3159432331/3+2601328μ8/3478296932/3+3598016μ314348907g1=1+2μ313+73μ323+169μ3+7627μ343+38281μ353+469μ32+58081μ373+97681μ383+10882729μ33+h1=1+4μ1/331/3+26μ2/3332/3+116μ27+161μ4/32731/3+740μ5/38132/3+3316μ2729+1540μ7/324331/3+2188μ8/324332/3+84460μ319683

where the coefficients 𝓝i1 are given appendix (D-L1)

Substituting back into equation (11), r2 is being a function of b1, d1, e1, f1, g1 and h1. It can be written in the form

r2=b1d1(n21+μ+b1+1μg1μe13μ2γ1γ22f1+31μ2σ1σ22h1)(12)

since the location of L1 is given by

ξ0,L1=1μ+r2(13)

from equation (12) and (13) we get

ξ0,L1=1i=39Ni1μ3i/3(14)

where the coefficients Ni1 are given appendix (B-L1)

4.1 Location of L2

The geometry of L2 can be visualized as given by Figure 2.

Follow the same procedure as with L1, with the corresponding values of𝓑1 = 1, 𝓑2 = −1. Substituting into (5) we get

r1r2=1,r1=x+μ,r2=x+μ1,r1x=r2x=1(15)

Substituting from (15) into (3) we get

Ux=n2(1μ+r2)(1μ)r12+3(1μ)(2σ1σ2)2r1415(1μ)(σ1σ2)2r16y2μr22+3μ2γ1γ22r2415μγ1γ22r26y2(16)

Then it may be reasonable in our case to assume that position of the equilibrium point L2 is the same as given by classical (RTBP) but perturbed due to the triaxial primaries

r1=a2+ε2,r2=b2+ε2,a2=1+b2.(17)

where a2 and b2 are the unperturbed positions of r1 and r2 respectively, and b2 is given after some successive approximation by the relation

b2=α1+α3α29227α3+281α4,α=μ3(1μ)3(18)

Substituting from equations (17) into equation (16) we get

Ux=n2(1μ+b2+ε2)1μ(1+b2)212ε21+b23(1μ)(2σ1σ2)2(1+b2)414ε21+b2+15(1μ)(σ1σ2)2(1+b2)616ε21+b2y2μb2212ε2b23μ(2γ1γ2)2b2414ε2b2+15μ(γ1γ2)2b2616ε2b2y2

Since y = 0, then the above equation may be written as

Ux=n2(1μ+b2+ε2)1μ(1+b2)212ε21+b23(1μ)(2σ1σ2)2(1+b2)414ε21+b2μb2212ε2b23μ(2γ1γ2)2b2414ε2b2=0(19)

Equation (19) can be solved for ε2 yielding

ε2=n2+2μb23+21μ1+b23+6μ2γ1γ2b25+61μ2σ1σ21+b251×n2(μb21)+μb22+1μ(1+b2)2+3μ(2γ1γ2)2b24+3(1μ)(2σ1σ2)2(1+b2)4(20)

Setting

d2=n2+2μb23+21μ1+b23+6μ2γ1γ2b25+61μ2σ1σ21+b251,e2=1b22,f2=1b24,g2=1(1+b2)2,h2=1(1+b2)4.

Using the above relations, equation (20) can be written in the form

ε=d2[n2(μb21)+(1μ)g2+μe2+9(1μ)A12h2+9μA22f2+3μ(2γ1γ2)2f2+3(1μ)(2σ1σ2)2h2](21)

where the coefficients 𝓝 are given appendix (D-L2) and where, d2, e2, f2, g2 and h2 are functions of μ andare given by

b2=μ313+13μ32319μ3+2527μ343+5681μ35313μ32+4627μ373+14581μ383μ33+d2=i=09Ni2μ3i/3e2=59233μ1/3+3μ2/3209μ31/3+5881μ32/3+10243μ3932729μ34/3+196243μ35/3+6736561μ323929619683μ37/3+3079619683μ38/3+42916177147μ33+f2=14327+3μ4/34μ+1493μ2/3148273μ1/31012243μ31/310538μ26561μ3245687253144μ37/3+3600328531441μ37/32258600531441μ32g2=12μ31/3+73μ32/3169μ33227μ34/3+34781μ35/3509μ325281μ37/3+1049μ38/312826729μ33+h2=14μ31/3+263μ32/311699μ3+899μ34/3+19627μ35/3294881μ32+146027μ37/353227μ38/364588729μ33

Substituting back into equation (21), r2 is a function of b2, d2, e2, f2, g2 and h2 and can be written in the form

r2=b2+d2[n2(μb21)+(1μ)g2+μe2+3μ(2γ1γ2)2f2+3(1μ)(2σ1σ2)2h2](22)

since the location of L2 is given by

ξ0,L2=1μ+r2(23)

from equation (22) and (23) we get

ξ0,L2=1i=39Ni2μ3i/3

where the coefficients Ni2 are given appendix (A-L2)

4.2 Location of L3

The geometry of L3 can be visualized as given by Figure 2.

Follow the same procedure as in L1, with the corresponding values of 𝓑1 = – 1, 𝓑2 = 1, substituting into (5) we get

r2r1=1,r1=(x+μ),r2=1μx,r1x=r2x=1(24)

Hence substituting from equation (24) into (5), we get

Ux=n2(1μr2)+(1μ)r12+3(1μ)(2σ1σ2)2r1415(1μ)(σ1σ2)2r16y2+μr22+3μ2γ1γ22r2415μγ1γ22r26(25)

Then it may be reasonable in our case to assume that position of the equilibrium point L3 is the same as given by classical restricted three-body problem but perturbed due tothe triaxial primaries

r1=a3+ε3,r2=b3+ε3,a3=b31.(26)

where a3 and b3 are the unperturbed values of r1 and r2 respectively, and b3 is given after some successive approximation by the relation

b3=27μ12161μ31728+(27)

Substituting from equations (26) into equation (25) and retaining terms up to the first order in the small quantities ε3 we get

Ux=n21μb3ε3+1μ(b31)212ε3b31+3(1μ)(2σ1σ2)2(b31)414ε3b3115(1μ)(σ1σ2)2(b31)616ε3b31y2+μb3212ε3b3+3μ(2γ1γ2)2b3414ε3b315μ(γ1γ2)2b3616ε3b3y2

Since y = 0, then the above equation may be written as

Ux=n21μb3ε3+1μ(b31)212ε2b31+3(1μ)(2σ1σ2)2(b31)414ε3b31+μb3212ε3b3+3μ(2γ1γ2)2b34(14ε3b3)=0(28)

Equation (28) can be solved for ε3 to yield

ε3=n221μb3132μb336μ2γ1γ2b3561μ2σ1σ2b3151×n2+n2μ1μ1+b32μb32+n2b33μ2γ1γ22b3431μ2σ1σ221+b34(29)

now, letting

d3=n22(1μ)b3132μb336μ(2γ1γ2)b356(1μ)(2σ1σ2)b315e3=1b32,f3=1b34,g3=1(b31)2,h3=1(b31)4

Using the above relations, equation (29) can be written in the form

ε3=d3(n2(1μ)+n2b3(1μ)g3μe33μ(2γ1γ2)2f33(1μ)(2σ1σ2)2h3)(30)

where d3, e3, f3, g3 and h3 are functions of μ. They are given by

d3=i=29Ni3μ3i/3e3=14+7μ48+49μ2768+665μ313824+f3=116+7μ96+245μ24608+2359μ355296+g3=19+7μ162+49μ23888+2135μ3209952+h3=181+7μ729+245μ252488+791μ3236196+

where the coefficients 𝓝i1 are given appendix (D-L3).

Substituting back into equation (29) to yield r2 as a function of b3, d3, e3, f3, g3 and h3. It can be rewritten in the form

r2=b3+d3(n2(1μ)+b3(1μ)g3μe33μ(2γ1γ2)2f33(1μ)(2σ1σ2)2h3)(31)

since the location of L3 is given by

ξ0,L3=1μr2(32)

from equation (31) and (32) we get

ξ0,L3=1i=39Ni3μi/3(33)

where the coefficients Ni3 are given Appendix (A-L3)

5 Numerical representations

In the following section, we will draw the locations of collinear points Li, i = 1, 2, 3 versus the mass ratio μ ∈ (0, 0.1) taking into account the effect oftrixial primaries.In all casesthe black curve represents the classical unperturbed RTBP in which the primaries are considered spheres. The blue and the red curves represent the increasing magnitude in the perturbing parameters.

5.1 Analysis of L1 location

Considering the massive primary as oblate, it can be seen in Figure 4a that the location of is shifted towards the massive primary, i.e. towards the barycenter. Since any equilibrium point emerges from the balance between the gravitational field and the rotational field of the primaries, we can conclude that the resultant of these forces is to perturb the location of L1 towards the massive primary. This is very logical dynamical effect, since the additional mass bulge due to oblateness of the massive primarycauses gravitational attraction towards the center. The effect is noticeable for a relatively higher mass ratio. In view of Figure 4b, and considering the less massive primary as oblate, the location of L1 is largely shifted towards the less massive primary, i.e. away from the barycenter. The effect of perturbation is much bigger than that in Figure 4a due to the close proximity of the point to the less massive primary.

The location of  against μ under the effect of an oblate less massive primary and sphere massive primary
Figure 4a

The location of against μ under the effect of an oblate less massive primary and sphere massive primary

The location of against μ under the effect of an oblate less massive primary and sphere massive primary
Figure 4b

The location of against μ under the effect of an oblate less massive primary and sphere massive primary

In view of the Figure 4c and Figure 4d, the dynamical effects are towards and away from the barycenter, respectively. The size of perturbation is small comparedto the effects of oblateness due to the magnitude of the perturbing parameter.

The location of L1 against μ under the effect of triaxial massive primary and sphere less massive primary
Figure 4c

The location of L1 against μ under the effect of triaxial massive primary and sphere less massive primary

The location of L1 against μ under the effect of an oblate massive primary and sphere less massive primary
Figure 4d

The location of L1 against μ under the effect of an oblate massive primary and sphere less massive primary

5.2 Analysis of L2 location

From Figure 4e, considering the massive primary as oblate, the location of L2 is shifted towards the barycenter for mass ratios larger that the classical critical mass ratio, namely the Routian value μc = 0.03841. In Figure 4f, considering the less massive primary as oblate, the location of L2 is also largely shifted towards to thebarycenter. The effect of perturbation is much bigger than that shown in Figure 4e due to the close proximity of the point to the less massive primary.

The location of L2 against μ under the effect of an oblate massive primary and sphere less massive primary
Figure 4e

The location of L2 against μ under the effect of an oblate massive primary and sphere less massive primary

The location of L2 against μ under the effect of an oblate less massive primary and sphere massive primary
Figure 4f

The location of L2 against μ under the effect of an oblate less massive primary and sphere massive primary

Referring to Figure 4g and Figure 4h, the dynamical effects are towards and away from the barycenter respectively. These effects could be easily interpreted by considering balancing between the gravitational and rotational fields. The size of perturbation is small comparedto the effects of oblateness due to the magnitude of the perturbing parameter.

The location of L2 against μ under the effect of triaxial massive primary and sphere less massive primary
Figure 4g

The location of L2 against μ under the effect of triaxial massive primary and sphere less massive primary

The location of L2 against μ under the effect of an oblate massive primary and sphere less massive primary
Figure 4h

The location of L2 against μ under the effect of an oblate massive primary and sphere less massive primary

5.3 Analysis of L3 location

The Figures 4i-4l show the locations of L3 considering the massive primary as oblate in 4i and the less massive primary as oblate in Figure 4j. Figure 4k and 4l show the perturbed location of L3 due to the triaxiality of the primaries. It is very clear that the dynamics are dominated by the massive primary.

The location of L3 against μ under the effect of an oblate massive primary and sphere less massive primary
Figure 4i

The location of L3 against μ under the effect of an oblate massive primary and sphere less massive primary

The location of L3 against μ under the effect of an oblate less massive primary and sphere massive primary
Figure 4j

The location of L3 against μ under the effect of an oblate less massive primary and sphere massive primary

The location of L3 against μ under the effect of triaxial massive primary and sphere less massive primary
Figure 4k

The location of L3 against μ under the effect of triaxial massive primary and sphere less massive primary

The location of L3 ratio μ under the effect of an oblate massive primary and sphere less massive primary
Figure 4l

The location of L3 ratio μ under the effect of an oblate massive primary and sphere less massive primary

6 Conclusion

We have treated the perturbed RTBP with the triaxiality of both primaries up to the terms of order 𝓞 1/r1,27 perturbations. As expected, these perturbations bring deviations of the locations of the equilibrium points from classical RTBP. In this work, we computed and illustrated these deviations in collinear points explicitly as functions in the mass ratio. We analyzed the oblate RTBP as special cases of a triaxialproblem. All the dynamical effects could be properly interpreted in view of balancing between the gravitational and rotational fields. It is observed that the dynamics of L3 are clearly dominated by the massive primary, while the dynamics of L1 and L2 are dominatedby the less massive primary due to their close proximity to it. When investigating the triaxiality effects, it is noticed that the size of perturbation is small compared to the effects of oblateness due to the magnitude of the perturbing parameter.

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Appendix (D-L1)

N01=2n24+305γ1305γ223σ1+3σ22+3μ1/375γ1+752γ2+3μ2/39γ192γ2N11=15+5n222515γ13+2515γ26+15σ1152σ2N21=59145n212+15865γ1915865γ21835σ1+35σ22N31=287318+335n2978860γ127+39430γ227+172σ1386σ23N41=288493154n2+301145γ181301145γ2163215σ13+215σ26N51=857518+108n2790940γ1243+395470γ2243+502σ19251σ29N81=462913729+26615n25431442824γ16561+15721412γ26561+704σ19352σ29N91=28068072187145255n2243+56532350γ15904928266175γ2590492320σ1243+1160σ2243

Appendix (D-L2)

N02=1n24305γ1+305γ223σ1+3σ22+3μ1/375γ1+752γ2+3μ2/39γ1+92γ2N12=125n222245γ13+2245γ2615σ1+152σ2N22=59145n21210465γ19+10465γ21835σ1+35σ22N32=299918335n2929450γ127+14725γ227118σ13+59σ23N42=27573335n2444105γ181+44105γ216335σ13+35σ26N52=2217554469n2380030γ1243+40015γ2243+128σ1964σ29N62=1231272433305n212690895γ1729+690895γ2145885σ127+85σ254N72=59750814360n293651725γ12187+3651725γ24374215σ19+215σ218N82=83107172944755n25412965156γ16561+6482578γ2656144σ19+22σ29N92=36518452187332240n224355392640γ159049+27696320γ2590492270σ1243+1135σ2243

Appendix (D-L3)

N03=1423n212σ1+6σ2N33=16484+560n218γ1+9γ2+2256σ11128σ2N63=71536(4032+10000n2810γ1+405γ2+34680σ117340σ2)N93=724576(384932+491120n265610γ1+32805γ2+1090560σ1545280σ2)

Appendix (A-L1)

N01=29n24+n44+494γ1805n2γ12+5139γ12247γ2+805n2γ24+75γ2σ22+5139γ224+3σ1+9n2σ14+150γ1σ175γ2σ19σ129σ2243σ229n2σ2875γ1σ2+9σ1σ25139γ1γ2N11=102γ1+3454n2γ11971γ12+51γ23458n2γ2+1971γ1γ219714γ2290γ1σ1+45γ2σ1+45γ1σ2452γ2σ2 N21=(9γ19n2γ1+567γ1292γ2+92n2γ2567γ1γ2+5674γ22+27γ1σ1272γ2σ1272γ1σ2+274γ2σ2N31=81γ12+81γ1γ281γ224N11=18+79n2411n444720γ13+15319n2γ11211556γ12+2360γ2315319n2γ224+11556γ1γ22079γ223σ1272n2σ1361γ1σ1N21=75268n23+44n43+70511γ11828270n2γ19+22125γ1270511γ236+14135n2γ2922125γ1γ2+221254γ222σ1+1534n2σ1+2155γ1σ13N31=407318+1121n231831n436217631γ127+337225n2γ15436168γ12+217631γ254337225n2γ2108+36168γ1γ29042γ22+9σ12327n2σ148372γ1σ19+4186γ2σ19+53σ129σ24+327n2σ28+4186γ1σ292093γ2σ2953σ1σ2+53σ224N41=327192006n23+4571n436+2253133γ11621630015n2γ1162+154879γ123154σ222253133γ2324+1630015n2γ2324154879γ1γ23+154879γ221255σ13+49n2σ1+15σ1σ2 N51=2824727+33547n22724991n41084893158γ1243+3138185n2γ1243572404γ129+2446579γ22433138185n2γ2486+572404γ1γ29143101γ229+641σ118797n2σ14130655γ1σ1108+130655γ2σ1162+12σ12641σ236+797n2σ28+130655γ1σ2162130655γ2σ232412σ1σ2+3σ22N61=787937486854033n2486+22696n481+35047541γ114589005530n2γ1729+15905525γ1224335047541γ22916+4502765n2γ272915905525γ1γ2234+15905525γ229728557σ1162+589n2σ13+589n2σ13+350242γ1σ1243175121γ2σ124378σ12+8557σ2324589n2σ218175121γ1σ2243+175121γ2σ2486+78σ1σ214σ22 N71=954265486+271853n21621273n41249027529γ12187+59331695n2γ1874812948929γ12243+49027529γ2437459331695n2γ217496+12948929γ1γ224312948929γ22972+2923σ154257n2σ12799070γ1σ1729+399535γ2σ1729+15σ122923σ2108+257n2σ24+399535γ1σ2729399535γ2σ21812215σ1σ2+15σ224N81=240245914581313483n2291611426n427+17992143γ113122+12927712n2γ165561+20836592γ12729177992143γ2262446463856n2γ26556120836592γ1γ2729+5209148γ2272914753σ1486+461n2σ1368007568γ1σ12187+4003784γ2σ121876σ12+14753σ2972461n2σ272+4003784γ1σ221872001892γ2σ22187+6σ1σ23σ222N91=11914273940255n22187+1084129n4972164982886γ119683626757695n2γ178732+223007333γ126561+82491443γ219683+626757695n2γ2157464223007333γ1γ26561+223007333γ2226244913σ11458+5333n2σ181+188329745γ1σ16561188329745γ2σ113122+913σ229165333n2σ2162188329745γ1σ213122+188329745γ2σ226244

Appendix (A-L2)

N02=13n24n44+479γ1727n2γ12+3681γ12479γ22+727n2γ243681γ1γ2+3681γ224+6σ19n2σ14+348γ1σ1174γ2σ13σ2+9n2σ28+9σ224174γ1σ2+87γ2σ29σ1σ2+9σ12N12=93γ13274n2γ1+1971γ12932γ2+3278n2γ21971γ1γ2+19714γ22+144γ1σ172γ2σ172γ1σ2+36γ2σ2N22=9γ19n2γ1+567γ1292γ2+92n2γ2567γ1γ2+5674γ22+27γ1σ1272γ2σ1272γ1σ2+274γ2σ2N32=81γ1281γ1γ2+81γ224N12=1241n2411n44+4354γ1311617n2γ112+82832/3γ122177γ23+11617n2γ2242484γ1γ2+621γ22+42σ1272n2σ1+283γ1σ1283γ2σ12+274n2σ2283γ1σ22+9σ1221σ2+283γ2σ249σ1σ2+94σ22 N22=71170n2344n43+48017γ11814182n2γ19182331/3γ1248017γ236+7091n2γ29+5469γ1γ254694γ22+110σ11534n2σ11649γ1σ13+1649γ2σ16+3σ1255σ2+1538n2σ2+1649γ1σ261649γ2σ2123σ1σ2+34σ22N32=4133186377n2361777n436+65279γ12768593n2γ15416752γ1265279γ254+68593n2γ2108+16752γ1γ24188γ22+757σ19201n2σ1416360γ1σ19+8180γ2σ1955σ12757σ212+201n2σ28+8180γ1σ294090γ2σ29+55σ1σ255σ224 N42=271161019n234229n436166043γ1162+63197n2γ116255721γ123+166043γ232463197n2γ2324+55721γ1γ2355721γ221244σ13n2σ150278γ1σ127+25139γ2σ12715σ12+22σ2+3n2σ22+25139γ1σ22725139γ2σ254+15σ1σ2154σ22N52=151152710601n22723497n41081449466γ1243+453517n2γ124327212γ129+724733γ2243453517n2γ2486+27212γ1γ296803γ2291951σ16+349n2σ14+15743γ1σ18115743γ2σ116212σ12+1951σ212349n2σ2815743γ1σ2162+15743γ2σ2324+12σ1σ23σ22 N62=177809486122333n248628960n48110534933γ11458+538370n2γ1729+3877025γ12243+10534933γ22916269185n2γ27293877025γ1γ2243+3877025γ2297262437σ1162+281n2σ13+563392γ1σ1243281696γ2σ1243+84σ12+62437σ2324281n2σ26281696γ1σ2243+140848γ2σ223484σ1σ2+21σ22N72=814116280365n248621469n4366513922γ1218722227281n2γ18748+4233653γ12243+3256961γ22187+22227281n2γ2174964233653γ1γ2243+4233653γ2297219063σ116257n2σ12+1420652γ1σ1729710326γ2σ1729+45σ12+19063σ2324+57n2σ24710326γ1σ2729+355163γ2σ272945σ1σ2+45σ224 N82=10010514581357369n2291627773n427+13081537γ11312229852018n2γ16561+3497408γ1272913081537γ236244+14926009n2γ265613497408γ1γ2729+874352γ22729+16265σ11624037n2σ13611987038γ1σ12187+5993519γ2σ12187+6σ1216265σ2324+4037n2σ272+5993519γ1σ221875993519γ2σ243746σ1σ2+3σ222N92=9693742187730099n27291680113n4972429534110γ1196831932138199n2γ173732269045573γ126561+214767055γ219683+1932138199n2γ2157464+269045573γ1γ26561269045573γ22262442665σ114584963n2σ181282835739γ1σ16561+282835739γ2σ113122+2665σ22916+4963n2σ2162+282835739γ1σ213122282835739γ2σ226244

Appendix (A-L3)

N03=2+182+3n2+12σ16σ2(2+2n26σ1+3σ2)N33=54164(84+560n218γ1+9γ2+2256σ11128σ2)1+n23σ1+3σ221128(23n212σ1+6σ2)(40+40n218γ1+9γ2384σ1+192σ2)N63=71536(4032+10000n2810γ1+405γ2+34680σ117340σ2)1+n23σ1+3σ22+637686γ13γ2+88σ144σ223n212σ1+6σ212048(84+560n218γ1+9γ2+2256σ11128σ2)×(40+40n218γ1+9γ2384σ1+192σ2)N93=749152(593128+20824n2+595488n4+174420γ117100n2γ110692γ1287210γ2+8550n2γ2+10692γ1γ22673γ221543128σ1+367008n2σ1+367308γ1σ1183654γ2σ1+1230336σ12+771564σ2183504n2σ2183654γ1σ2+91827γ2σ21230336σ1σ2+307584σ22)

About the article

Received: 2018-01-11

Accepted: 2018-04-22

Published Online: 2018-09-06


Citation Information: Open Physics, Volume 16, Issue 1, Pages 525–538, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2018-0069.

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© 2018 S. Z. Alamri et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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