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The effect of zonal harmonic coefficients in the
framework of the restricted three–body problem
Elbaz. I. Abouelmagd∗
a. Mathematics Department, Faculty of Science and Arts (Khulais), King Abdulaziz,University, Jeddah, Saudi Arabia.
b. Nonlinear Analysis and Applied Mathematics Research Group (NAAM) Department ofMathematics, King Abdulaziz University, Jeddah, Saudi Arabia.
M. S. Alhothuali1, Juan L.G. Guirao2
Departamento de Matematica Aplicada y Estadıstica. Universidad Politecnica deCartagena, Hospital de Marina, 30203–Cartagena, Region de Murcia, Spain.
H. M. Malaikah3
Abstract
The objective of this paper is to present a comprehensive analytical study onthe existence of the libration points and their linear stability in the frame ofthe restricted three–body problem considering the effect of the first two evenzonal harmonics parameters with respect to both primaries. Moreover, theperiodic orbits around the libration points, the expressions for semi–majorand semi–minor axes, the eccentricities and the periods of elliptical orbitsas well as the orientation of the principal axes are stated. In addition, wesupport our study with some numerical and graphical experiments.
Keywords: Restricted three-body problem; Libration points; Stability;
The effect of zonal harmonic coefficients in the framework of the restricted three–bodyproblem
∗Corresponding authorEmail addresses: [email protected], [email protected] (Elbaz. I.
Abouelmagd), [email protected] (Juan L.G. Guirao)1Same address than corresponding author, affiliation b.2Web page: http://www.jlguirao.es3Same address than corresponding author, affiliation b.
Preprint submitted to Advances in Space Research November 23, 2014
Zonal harmonics coefficients; Periodic orbits.
1. Introduction
The three–body problem is one of the central problems in field of CelestialMechanics. It has many applications in different scientific areas, in partic-ular, in the fields of Astrophysicsics and astrodynamics. This problem isclassified in two classes: the first one is the general problem which describesthe motion of three celestial bodies under their mutual gravitational attrac-tion. The second class is the restricted problem in which the third bodyhas an infinitesimal mass compared with masses of the other two bodies andconsequently it does not affect their motion.
One of the main applications of the general problem in Astrophysicsicsis for instance the dynamics of triple stars systems. In the second half ofthe 20th century and even today. the study of scientific community hascenter its attention on the restricted three–body problem and there are a bignumber of papers studying different aspect of this problem. For instance,considering the influences of perturbed forces such as oblateness, radiationpressure, Coriolis and centrifugal forces, variation of masses, the Pointing–Robertson effect, the atmospheric drag, the solar wind,..., etc.
Significant studies related with the libration points considering the oblate-ness of one or both primaries when the equatorial plane is coincident with theplane of motion are done by Subbarao&Sharma (1975), Sharma&Subbarao(1978) and Markellos et al. (1996).
Some works studying different aspects of the dynamics of the restrictedproblem when the three participating bodies are oblate spheroids are givenby El-Shaboury&El-Tantawy (1993), Abouelmagd&El-Shaboury (2012) andElipe & Ferrer (1985). Interesting papers when one or both primaries are tri-axial bodies are for instance El-Shaboury et al. (1991), Khanna&Bhatnagar(1999) and Sharma&Bhatnagar (2001)
Several authors have been devoted their efforts to study the effects ofsmall perturbations in centrifugal and Coriolis forces as Szebehely (1967),Bhatnagar&Hallan (1978), Devi&Singh (1994) and Shu et al. (2005).
The existence and the linear stability of the libration points in the re-stricted problem for perturbed potentials between the bodies in the cases: thebigger primary is an oblate spheroid, both primaries are oblate spheroids, the
2
primaries are spherical and the biggest primary is a source of radiation werestudied in by Bhatnagar&Hallan (1979). They observed that the collinearpoints are unstable, the range of stability for the triangular points increasesor decreases depending on the sign of a parameter which depends on theperturbed functions. Furthermore, Singh&Ishwar (1999) studies the effectof the oblateness and the radiation pressure at the locations of the triangu-lar points and their linear stability when both the primaries are oblate andradiating.
Ishwar&Elipe (2001) found the secular solutions at the triangular pointsin the generalized photogravitational restricted three–body problem. Theproblem is generalized in the sense that the bigger primary is a source ofradiation and the smaller one is an oblate spheroid. Moreover, Abouelmagdet al. (2014a) found the secular solution around the triangular equilibriumpoints and reduce it to a periodic solution in the frame work of the generalizedrestricted thee–body problem, in sense that both primaries are oblate andradiating as well as the gravitational potential from a belt. They also showedthat the linearized equation of motion of the infinitesimal body around thetriangular equilibrium points has a secular solution when the value of themasses ratio is equal to the critical mass value. Numerical and graphicalanalysis in order to understand the effects of the perturbed forces are stated.
Mittal&Bhatnagar (2009) studies the periodic orbits generated by La-grangian solutions of the restricted three–body problem when the biggerbody is a source of radiation and the smaller is an oblate spheroid. It isused the definition of Karimov&Sokolsky (1989) for mobile coordinates todetermine these orbits and the predictor method to draw them.
Singh&Begha (2011) studied the existence of periodic orbits around thetriangular points in the restricted three-body problem when the bigger pri-mary is triaxial and the smaller one is considered as an oblate spheroid. Inthe range of linear stability under the effects of the perturbed forces of Cori-olis and centrifugal, it is deduced that long and short periodic orbits existaround these points and are stated their periods, orientation and eccentrici-ties affected by the non sphericity and the perturbations in the Coriolis andcentrifugal forces.
Abouelmagd (2012) studies the effects of oblateness coefficients J2 and J4
of the bigger primary in the planar restricted three–body problem on thelocations of the triangular points and their linear stability. It was foundthat these locations are affected by the coefficients of oblateness. Further-
3
more, it was stated that the triangular points are stable for 0 < µ < µc
and unstable when µc ≤ µ ≤ 1/2, where µc is the critical mass parameterwhich depends on the coefficients of oblateness. Some numerical values forthe positions of the triangular points for certain planets of the solar systemwere stated remarking that the range of stability decrease. Finally, someexamples showing that there is no influence for J4 on the range of stabil-ity for some planets systems as Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems. Furthermore, the existence of new equilibrium points forthe restricted three–body problem with equal prolate primaries is stated inDoukos et al. (2012). It was found that these points are located on the Z–axis above and below the inner Eulerian equilibrium point L1 as well as anew type of straight–line periodic oscillations, different from the well–knownSitnikov motions. The stability properties of these oscillations are used tofind new types of families of 3D periodic orbits branch out of the Z–axisconsisting of orbits located entirely above or below the orbital plane of theprimaries. Recently, there are also some interesting papers connected withthe restricted three–body problem, see Beevi&Sharma (2012), Singh (2012),Singh&Taura (2013), Kishor&Kushvah (2013), Abouelmagd et al. (2013),Abouelmagd (2013) and Abouelmagd et al. (2014b).
In last decades a great number of authors have studied the restrictedthree–body problem taking account the effects of oblateness of one or bothprimaries up to 10−3 of the main terms of the potential. But our maincontribution and major modification in the present work is to study theeffect of oblateness up to 10−6 when both primaries are oblate spheroids.Therefore we shall consider the influence of even zonal harmonic parametersup to J4 for both primaries on the existence of the libration points and theirlinear stability as well as we shall analyze the existence of periodic orbitsaround these points.
This work is organized as follow: a historical review on the importanceof the three–body problem and the aim of the present work is presented incurrent Section. The mean motion and equations of motion of the problemunder consideration are derived in Sections 2 and 3 . In Section 4 we findthe locations of the libration points and in Section 5 we present a studyof their linear stability. In Section 6 is obtained the expression of criticalmass and in Section 7 this notion is extended to find periodic orbits aroundlibration points. Finally, some pictures showing the conclusions of our studyare presented in the last section. We underline that the model studied in
4
this work has special importance in space missions either to send telescopesor for dispatching satellites or exploring vehicles to stable regions to move ingravitational fields of planets systems.
2. Mean motion
We assume that Ai = J2iR2i1 and Bi = J2iR
2i2 represent the oblateness
coefficients of the bigger and smaller primaries respectively where J2i andJ2i, (i = 1, 2) give the zonal harmonics coefficients that characterize the sizeof non–spherical components of the potential with respect to the primarieswhile R1 and R2 are the mean radius of the primaries.
From the potential theory, if the body has axial symmetry with mass m0
and mean radius R0, then the external gravitational potential of this bodycan be written as
V = −Gm0
r
[1−
+∞∑n=2
Jn
(R0
r
)pn(sin(δ))
](1)
where r is the radial distance from the center of the object to the center ofany other body and Jn is a dimensionless coefficient that characterizes thesize of the non–spherical components of the potential, when n is even Jn arecalled the zonal harmonic coefficients, pn[sin(δ)] are the Legendre polynomialsof degree n and δ denotes the latitude of the body, see Murray&Dermott(1999) for more details. In this work, we study the planar problem whenthe infinitesimal body moves in the plane of motion of the primaries withouteffects on their motion. Therefore, we assume this plane coincides with theequatorial plane of the bigger massive primary, then δ = 0. ConsequentlyEq. (1) with the effect of oblateness up to J4 is reduced to the followingequation:
V = −Gm0
[1
r+
J2R20
2r3− 3J4R
40
8r5
], (2)
for more details see Abouelmagd (2012).
Now let the masses of the bigger and smaller primaries be m1 and m2
respectively. Let r = ui + vj be the position vector of m2 with respect tom1. In this setting, Eq. (2) implies that the potential V1 between m1 andm2 has the form:
5
V1 = −Gm1m2
[1
r+
(A1 + B1)
2r3− 3(A2 + B2)
8r5
].
If both bodies revolve under mutual gravitational attraction in circularorbits on their center of mass, the equation motion for m2 is
r = −(
m1 + m2
m1m2
)∇V1, (3)
where ∇ = i∂
∂u+ j
∂
∂v.
Note that Eq. (3) has as a particular solution:
r = d, u = r cos ϕ, v = r sin ϕ, ϕ = n, (4)
where d is constant while n is the called mean motion and can be written as:
n2 = G(m1 + m2)
[1
r3+
3(A1 + B1)
2r5− 15(A2 + B2)
8r7
]. (5)
Remark 1. The relation of Eq. (5) agrees with the result of Abouelmagd&El-Shaboury (2012) when are not considered the effects of J4 and J4 (A2 = B2 =0). Moreover, note that when the bigger primary is oblate and the smaller isconsidered as a point mass (B1 = B2 = 0), then Eq. (5) is the same equationpreviously stated in Abouelmagd (2012).
3. Equations of motion
Let b1, b2 and b be three bodies located at (X1, Y1, Z1), (X2, Y2, Z2) and(X, Y, Z) in a sidereal frame, their masses are m1, m2 and m respectivelybeing m1 and m2 the primaries masses having a circular orbit around theircommon center of mass. m is the mass of the infinitesimal body that movesin the same motion plane of the primaries under their gravitational fieldwithout affecting their motion.
We assume that G(m1 + m2) = 1 where the sum of primaries masses isone. Hence the gravitational constant G becomes also one. If the mutualdistance r between the primaries is taken one, we deduce that the unper-turbed mean motion is also equal to one, because G(m1 + m2) = n2r3, seeMurray&Dermott (1999) for more details. Now, choosing synodic coordi-nates which rotate with angular velocity n in positive direction and have the
6
Figure 1: The relation between the inertial XY and rotating xy coordinates systems inthe restricted three–body problem.
same origin of the sidereal reference frame at the common center of mass,consequently we have m1 = 1 − µ and m2 = µ ≤ 1/2. In this setting thecoordinates of the three bodies b1,b2 and b can be written in a synodic frameas (µ, 0, 0), (µ− 1, 0, 0) and (x, y, z) respectively, see Figure 1.
According to our assumptions the equations of motion in a synodic coor-dinate system xy with dimensionless variables are controlled by
x− 2ny = Ωx,
y + 2nx = Ωy,(6)
where
Ω =1
2n2[(1− µ)r2
1 + µr22] + (1− µ)[
1
r1
+A1
2r31
− 3A2
8r51
] + µ[1
r2
+B1
2r32
− 3B2
8r52
],
r21 = (x− µ) + y2, r2
2 = (x− µ + 1)2 + y2,m1 = (1− µ), m2 = µ ≤ 1/2, m1 ≥ m2,
n2 = 1 +3
2(A1 + B1)−
15
8(A2 + B2). (7)
7
4. Positions of the libration points
Since Ω = Ω(x, y), then
dΩ
dt= xΩx + yΩy. (8)
Now, from Eqs. (6) and (8) the Jacobi integral of the system can be writtenas
x2 + y2 − 2Ω + c = 0.
The locations of the equilibrium points Li, i = 1, ..., 5 be determined by thesolutions of
Ωx = Ωy = 0 (9)
where
Ωx = (x− µ)f1(r1) + (x− µ + 1)f2(r2),
Ωy = y(f1(r1) + f2(r2)).(10)
being
f1(r1) = (1− µ)
[n2 −
(1
r31
+3A1
2r51
− 15A2
8r71
)],
f2(r2) = µ
[n2 −
(1
r32
+3B1
2r52
− 15B2
8r72
)].
(11)
Eqs. (9) and (10) show that there are two cases to analyze that we shalltreat in the next subsections.
4.1. Location of the collinear points
The location of the collinear points Li, i = 1, 2, 3 is determined by Ωx =Ωy = 0 and y = 0. By Eqs. (9) and (10), this property is translated in
x
[1 +
3
2(A1 + B1)−
15
8(A2 + B2)
]−(1− µ)(x− µ)
(1
|x− µ|3+
3A1
2|x− µ|5− 15A2
8|x− µ|7
)−µ(x− µ + 1)
(1
|x− µ + 1|3+
3B1
2|x− µ + 1|5− 15B2
8|x− µ + 1|7
)
= 0
(12)
8
Thus, the location of Li are respectively xi, i = 1, 2, 3, where
x1 = µ− 1− ξ1,x2 = µ− 1 + ξ2,x3 = µ + ξ3.
(13)
Now, substituting Eqs. (13) in Eq. (12) we obtain three polynomials ofdegree thirteenth in ξ1, ξ2 and ξ3 respectively, see the Appendix for theconcrete expressions. Solving these equations for the parameter of the massratio µ, we can express µ a series in ξi at the point Li, (i = 1, 2, 3), again seethe Appendix for the details.
From practical reasons we need to state the positions of the collinear pointsdepending only of one parameter. So, we use the Lagranges inversion methodto invert the series which represent µ to express ξi, i = 1, 2, 3, as functions ofµ, see the Appendix for the concrete form of these expressions. Consequentlywe obtain appropriate approximations for the locations of collinear points inthe form:
• At the point L1:
x1 =
µ− 1−
(15B2
8− 8s + 12A1 − 15A2
)1/6
µ1/6
−(8 + 4s + 24A1 − 45A2)3√
45B2
[3(8− 8s + 12A1 − 15A2)]4/3µ1/3
(14)
• At the point L2:
x2 =
µ− 1 +
(15B2
8− 8s + 12A1 − 15A2
)1/6
µ1/6
+(8 + 4s + 24A1 − 45A2)
3√
45B2
[3(8− 8s + 12A1 − 15A2)]4/3µ1/3
(15)
• At the point L3:
x3 =
µ +
(15A2
8− 8s + 12B1 − 15B2
)1/6
(1− µ)1/6
+(8 + 4s + 24B1 − 45B2)
3√
45A2
[3(8− 8s + 12B1 − 15B2)]4/3(1− µ)1/3
(16)
9
where s = n2
The accuracy of approximations according to the value of mass ratio inEqs. (14), (15) and (16) is not very high when the oblateness parameters arevery small but there are a lot of technical difficulties to obtain more accurateresults computing the effects of all oblateness parameters. In order to improvethis we have considered zero the parameters of oblateness A2, B1, B2 andA1 6= 0. Consequently, the positions of the collinear points under theseconditions are approximate by
• At the point L1:
x1 =
µ− 1− 1
3√
3(1− 5
6A1)µ
1/3 − 1
3 3√
9(1 +
5
6A1)µ
2/3
+1
27(1 +
5
2A1)µ−
50
243 3√
3(1− 73
300A1)µ
4/3
− 344
729 3√
9(1 +
1127
258A1)µ
5/3 +4
81(1 +
5
2A1)µ
2
(17)
• At the point L2:
x2 =
µ− 1 +
13√
3(1− 5
6A1)µ
1/3 − 1
3 3√
9(1 +
5
6A1)µ
2/3
+1
27(1 +
5
2A1)µ +
58
243 3√
3(1− 143
348A1)µ
4/3
− 11
729 3√
9(1 +
763
66A1)µ
5/3 − 4
81(1 + 2A1)µ
2
(18)
• At the point L3:
x3 =
µ + (1− µ)1/4 +
(2 + A1)
4A1
(1− µ)1/2
+(21 + 37A1)
24A21
(1− µ)3/4 − 2(3− 8A1)
3A31
(1− µ)
+3(221 + 794A1)
128A41
(1− µ)5/4 +112(198 + 895A1)
1536A51
(1− µ)3/2
(19)
Note that Eqs. (17), (18) and (19) represent the positions of collinearpoints with a better approximation in function of the mass ratio µ when
10
A2, B1, B2 and A1 6= 0, but Eq. (19) has a singularity when A1 = 0. Toovercome this, we have to consider A1 = 0 early in Eq. (12). However wecan also remove this singularity if we set A1 = 1− δ where 0 < δ < 1. HenceEq. (19) can be rewritten in the form
x3 =
µ + (1− µ)1/4 +
3
4(1 +
2
3δ)(1− µ)1/2
+29
12(1 +
79
58δ)(1− µ)3/4 +
10
3(1 +
7
5δ)(1− µ)
+3045
128(1 +
3266
1015δ)(1− µ)5/4 +
7651
96(1 +
4570
1039δ)(1− µ)3/2
(20)
4.2. Location of the triangular points
The locations of the triangular points are given by Ωx = Ωy = 0 andy 6= 0. If Eq. (10) is satisfied, then f1(r1) = f2(r2) = 0 and we obtain
n2 =1
r31
+3A1
2r51
− 15A2
8r71
n2 =1
r31
+3B1
2r52
− 15B2
8r72
.(21)
The exact solutions of the triangular points L4,5 will be governed by
x4,5 = −1
2[1− 2µ + r2
1 − r22]
y4,5 = ±
√r21 + r2
2
2−(
r21 − r2
2
2
)2
− 1
4.
(22)
when the primaries are spherical bodies (Ai = Bi = 0, i = 1, 2), the solutionof Eqs. (21) are r1 = r2 = 1. Therefore, when both primaries are oblate wecan assume that the solutions of Eqs. (21) have the form
r1 = 1 + π1,r2 = 1 + π2
(23)
where π1, π2 1.
To have the appropriate approximation for the values of π1 and π2 we havesubstitute Eqs. (23) into Eqs. (21) with the help of Eqs. (7) such expression
11
of the value of n2 is given. Using binomial expansions is possible to obtain aseries in πi. Consequently, we get a quadratic equation in πi.
Since Ai = J2iR2i1 , Bi = J2iR
2i2 , then A2
1, B21 , A2, B2 and A1B1 are propor-
tional to the fourth powers of the radii of the primaries bodies. Hence, we canrestrict ourselves to the terms of order πi and π2
i and ignore terms of thirdorder or more. Solving this equation and joining together the expressionswhich contains Ai, Bi, A2
i , B2i , A1B1 (i = 1, 2), we have
π1 = −1
2B1 +
5
8B2 +
1
2B2
1 +5
4A1B1,
π2 = −1
2A1 +
5
8A2 +
1
2A2
1 +5
4A1B1.
(24)
Now, substituting Eqs. (24) into Eqs. (23) we get
r1 = 1− 1
2B1 +
1
8(5B2 + 4B2
1 + 10A1B1) ,
r2 = 1− 1
2A1 +
1
8(5A2 + 4A2
1 + 10A1B1) .
(25)
Substituting Eqs. (25) into Eqs. (22) we obtain
x4,5 = µ− 1
2− 1
2(A1 −B1) +
5
8(A2 −B2) +
5
8(A2
1 −B21) ,
y4,5 = ±√
3
2
1− 1
3(A1 −B1) +
5
12(A2 + B2)
+1
36[7(A2
1 + B21) + 68A1B1]
.
(26)
Remark 2. Note that the results obtained in Eqs. (26) are consistent with:
1. Classical problem, see the book Szebehely (1967) (Ai = Bi = 0, i = 1, 2.)
2. The results of Subbarao&Sharma (1975) (A1 6= 0, A2 = Bi = 0, i =1, 2.).
3. The results of Bhatnagar&Hallan (1979) (A1 6= 0, B1 6= 0, A2 = B2 =0).
4. The results of Abouelmagd (2012) (A1 6= 0, A2 6= 0, Bi = 0, i = 1, 2).
12
Table 1. Coordinates of the triangular points.
13
Note 1. Equations Eqs. (26) illustrate that the locations of the triangularpoints could change depending on the effects of the perturbed forces repre-sented by the sum terms and these elements could be zero or not. We numer-ically analyze the possible changes on these terms in Table 1 where x4,5 andy4,5 represent the coordinates of triangular points when A2 = B2 = 0 and x4,5
and y4,5 include all oblateness effects.
5. Stability of the libration points
Our physical problem is described by a continuous dynamical system withfour degrees of freedom given by x = f(x) with solution a function of thetime being x ∈ R4 and f = (f1, f2, f3, f4) a vector self–map from R4 intoitself.
In this setting the linearized solution of this system can be written as
x(t) =4∑
i=1
kivieλit
where vi ∈ R4 is the eigenvector associated to the eigenvalue λi and ki
is a constant that will be determined by the initial conditions where (i =1, 2, 3, 4), see Abouelmagd&El-Shaboury (2012) for more details.
Now, we assume that the variation ξ and η describe the possible motion ofthe infinitesimal mass in the vicinity of one libration points and this variationis defined as
ξ = x− x0,η = y − y0.
(27)
Substituting Eqs. (27) into Eqs. (6) we get the following two variationalequations:
ξ − 2nη = Ω0xxξ + Ω0
xyη,
η + 2nξ = Ω0xyξ + Ω0
yyη,(28)
here the partial derivatives of order two of Ω are denoted by the subscriptsx, y and the superscript 0 indicates that such derivative is evaluated at oneof the libration points. Since we are interested in the linear stability we only
14
shall consider the linear terms of ξ and η. Thus, the characteristic equationassociated to Eqs. (28) is
λ4 +(4n2 − Ω0
xx − Ω0yy
)λ2 + Ω0
xxΩ0yy −
(Ω0
xy
)= 0. (29)
5.1. Stability of the collinear points
Since y = 0 at the collinear points being
Ω0xx = f1(r1) + f2(r2) + r1
∂f1
∂r1
+ r2∂f2
∂r2
,
Ω0xy = 0,
Ω0yy = f1(r1) + f2(r2)
where f1(r1) and f2(r2) are stated in Eqs. (11) and
∂f1
∂r1
= (1− µ)
[3
r41
+15A1
2r61
− 105A2
8r81
]∂f2
∂r2
= µ
[3
r42
+15B1
2r62
− 105B2
8r82
].
(30)
Therefore,
Ω0xx =
1 +
3
2(A1 + B1)−
15
8(A2 + B2)
+(1− µ)
(2
r31
+6A1
r51
− 45A2
4r71
)+µ
(2
r32
+6B1
r52
− 45B2
4r72
)
,
Ω0xy = 0 and
Ω0yy =
1 +
3
2(A1 + B1)−
15
8(A2 + B2)
−(1− µ)
(1
r31
+3A1
2r51
− 15A2
8r71
)−µ
(1
r32
+3B1
2r52
− 15B2
8r72
)
.
Since the coordinates of L1 are (µ − 1 − ξ1, 0), then r1 = 1 + ξ1 andr2 = ξ1 with 0 < ξ1 1. Therefore, is possible to express Ω0
xx = f(ξ1) andΩ0
yy = g(ξ1) for certain functions f and g such that f(ξ1) ∼= f(0+) = ∞ andg(ξ1) ∼= g(0+) = −∞. Since Ω0
xxΩ0yy < 0, then Ω0
xxΩ0yy − (Ω0
xy)2 < 0 at L1. In
15
a similar way, it is possible to state that Ω0xxΩ
0yy − (Ω0
xy)2 < 0 for L2 and L3.
Then Eq. (29) can be written as
λ4 +(4n2 − Ω0
xx − Ω0yy
)λ2 + Ω0
xxΩ0yy = 0 (31)
where
λ2 = −1
2
(4n2 − Ω0
xx − Ω0yy
)±√(
4n2 − Ω0xx − Ω0
yy
)2 − 4Ω0xxΩ
0yy
. (32)
From the previous discussion the discriminant is positive and its square root isgreater than the absolute value for the coefficient of λ2 in Eq. (31). ThereforeEq. (32) gives two different values for λ2 one positive and other negative.Hence the four roots of Eq. (31) can be written as λ1,2 = ±σ, λ3,4 = ±iτwhere σ, τ are reals and i is the imaginary unit, then the general solution ofEq. (28) is represented by
ξ(t) =∑4
i=1 δieλit,
η(t) =∑4
i=1 ρieλit.
(33)
Remark 3. Note that although λ3,4 are pure imaginary the motion in prox-imity of collinear points is unbounded because λ1,2 are real and therefore thebehavior of stability of the collinear points does not change due to the oblate-ness perturbation and these points are unstables.
16
5.2. Stability of the triangular points.
In the case of triangular points we have f1(r1) = f2(r2) = 0, hence weobtain
Ω0xx =
1 +1
2(9A1 + B1)−
5
8(11A2 + B2)
+1
4(14A2
1 −B21) +
1
2A1B1
−µ
[4(A1 −B1)−
25
4(A2 −B2) +
15
4(A2
1 −B21)
]
Ω0xy = ±3
√3
4
1 +1
6(19A1 + 7B1)−
5
24(25A2 + 7B2)
+1
36(10A2
1 − 11B21) +
49
18A1B1
−µ
2 +1
3(4A1 + 13B1)−
20
3(A2 + B2)
+1
36(35A2
1 −B21) +
59
9A1B1
Ω0yy =
9
4
1 +11
6(A1 + B1)−
5
24(17A2 + 11B2)
− 1
12(14A2
1 −B21) +
19
6A1B1
+µ
[5
4(A2 −B2) +
5
4(A2
1 −B21)
]
.
(34)
From Eqs. (7) and (34) we get
17
4n2 − Ω0xx − Ω0
yy =
1− 3
2(A1 −B1) +
15
8(3A2 −B2)−
15
2A1B1
µ
[3(A1 −B1)−
15
2(A2 −B2)
]
Ω0xxΩ
0yy −
(Ω0
xy
)2=
27
4µ(1− µ)
1 +13
3(A1 + B1)
−20
3(A2 + B2)
14
3(A2
1 + B21) +
95
6A1B1
.
Now, let Λ2 = λ hence Eq. (29) becomes
Λ2 +(4n2 − Ω0
xx − Ω0yy
)Λ + Ω0
xxΩ0yy −
(Ω0
xy
)2= 0. (35)
The two roots of the previous equation are
Λ1,2 = −1
2
[C ±
√D]
(36)
where
C =
1− 3
2(A1 −B1) +
15
8(3A2 −B2)−
15
2A1B1
µ
[3(A1 −B1)−
15
2(A2 −B2)
] (37)
and D is the discriminant of Eq. (35) that can be written as
D = f(µ) = αµ2 + βµ + γ (38)
18
where the coefficients α, β and γ are given by
α = 27
1 +
13
3(A1 + B1)−
20
3(A2 + B2)
+5 (A21 + B2
1) +91
6A1B1
β = −27
1 +
1
9(37A1 + 41B1)−
5
9(11A2 + 13B2)
+5 (A21 + B2
1) +91
6A1B1
γ = 1− 3(A1 −B1) +
15
4(3A2 −B2) +
9
4(A2
1 + B21)−
39
2A1B1.
(39)
If the values Λ1,2 are negative and distinct, then the four roots of (29)
are pure imaginary and therefore we have stability. Note thatdf
dµ≤ 0 for
µ ≤ −β/2α and if µ = −β/2α < 1/2 we have that f(0) = γ and f(1/2) < 0,therefore by the Bolzano’s theorem plus f decreasing in the interval (0, 1/2),so there exists a unique value µc ∈ (0, 1/2) making D equals to zero.
The fact |Ai|, |Bi| 1 implies C > 0. Now, there are three possiblesituations. The first one µ = µc and D = 0 and the second one µc ≤ µ ≤ 1/2and D < 0 lead to unstable triangular points. The third and last possibilityis 0 < µ < µc and D > 0. Since C > 0, the values of Λ2 given by Eq. (36)will be negative and therefore the four roots of the characteristic equationare pure imaginary and different. Therefore the motion in the proximity ofthe triangular points is stable.
Remark 4. The effects of oblateness parameters will be graphically inves-tigated through the Figures 2–7 when µ = 0.01, µ = 0.02, µ = 0.038 andµ = 0.0384 obtaining the following conclusion: we observe that the discrim-inant value is negative when µ = 0.0384, therefore the value of the criticalmass must be less than 0.0384.
6. Critical mass
The discriminant D given by Eq. (38) is zero for
µc = − 1
2α
[β +
√β2 − 4αγ
].
19
Figure 2: Discriminant versus A1 and B1 when µ = 0.01 and A2 = B2 = 0.
Figure 3: Discriminant versus A2 and B2 when µ = 0.01 and A1 = 0.001 and B1 = 0.0005.
20
Figure 4: Discriminant versus A1 and B1 when µ = 0.02 and A2 = B2 = 0.
Figure 5: Discriminant versus A2 and B2 when µ = 0.02 and A1 = 0.001 and B1 = 0.0005.
21
Figure 6: Discriminant versus A2 and B2 when µ = 0.038 and A1 = 0.001 and B1 = 0.0005.
Figure 7: Discriminant versus A2 and B2 when µ = 0.0384 and A1 = 0.001 and B1 =0.0005.
22
The value µc can be written as
µc = µc0 + µc1 + µc2
where
µc0 =1
2
(1−
√69
9
),
µc1 = −1
9
(1 +
13√69
)A1 +
1
9
(1− 13√
69
)B1 and
µc2 =
5
18
(1 +
25
2√
69
)A2 −
5
18
(1− 25
2√
69
)B2
+13
27
(A2
1 −B21)
+9
1196√
69[1519(A2
1 + B21)− 576A1B1]
.
The value of critical mass when oblateness is ignored is µc0 while µc1
represents the terms involving the coefficients of oblateness J2 and J2 andµc2 includes some terms involving the effects of J4 and J4 and other termsproportional to second order terms of J2 and J2. In Table 2, µc representsthe critical mass value considering only the effects of J2 and J2. While µc
includes the effects of coefficients J2, J2, J4 and J4.
Remark 5. It is clear that the critical mass is affected by the even zonalharmonic coefficients because its value may be decrease or increase with theinfluence of J4 and J4 respect from the value obtain only considering theeffect of J2 and J2. However the value of critical mass in the classical casecould be bigger than the value obtained under the effect of these oblatenessparameters. The data calculated in Table 2 for critical mass supported thegraphical results obtained in the previous section where were stated that thecritical mass value must be less than 0.0384.
23
Table 2. The values of critical mass.
7. Periodic orbits around libration points
7.1. Motion in the proximity of the collinear points
The aim of this section is to obtain the periodic orbits existing aroundthe collinear points. Although one of the roots of characteristic polynomialof Eq. (31) is a real positive number and the motion is unbounded. Aftersubstituting Eqs. (33) into Eqs. (28) we get
ρi = niδi,
ni =λ2
i − Ω0xx
2nλi
=2nλi
Ω0yy − λ2
i
.(40)
Let ξ0, η0, ξ0 and η0 be the initial conditions evaluated at t = t0 associ-ated to Eqs. (28). Note that Eqs. (40) shows that the constants ρi and δi
(i = 1, 2, 3, 4) are dependent and determined for the initial conditions. Sub-stituting the first equation of Eqs. (40) into Eqs. (33) these conditions can
24
be written as
ξ0 =4∑
i=1
δieλit, η0 =
4∑i=1
niδieλit,
ξ0 =4∑
i=1
λiδieλit, η0 =
4∑i=1
niλiδieλit.
(41)
Since the determinant of the system of Eqs. (41) does not equal zero, because
−√
Ω0xx/Ω
0yy
[(Ω0
xx + Ω0yy − 4n2
)2 − 1
2Ω0
xxΩ0yy
]2
6= 0,
Therefore the system of Eqs. (41) can be inverted for obtaining the coeffi-cients as functions of the initial conditions.
Since the coefficients δ1 and δ2 are connected with the real exponents ofλ1 and and λ2 where one of them is negative and the other is positive. So thesolution will be unbounded. Without loss of generality we can choose initialconditions ξ0 and η0 such that δ1 = δ2 = 0. Thus, the solution only containstrigonometric functions and Eqs. (33) can be written as
ξ = ξ0 cos τ(t− t0) +η0
ksin τ(t− t0),
η = η0 cos τ(t− t0)ξ0
ksin τ(t− t0),
n3 = ik
(42)
where
k =τ 2 + Ω0
xx
2nτ=
2nτ
τ 2 + Ω0yy
and
τ =
1
2
[ √(4n2 − Ω0
xx − Ω0yy
)2 − 4Ω0xxΩ
0yy
−Ω0xx − Ω0
yy + 4n2
]1/2
.
From Eqs. (42), we obtain
ξ0 =η0τ
k,
η0 = −ξ0kτ.(43)
Eqs. (43) shows that the election of the initial conditions ξ0 and η0 deter-mines the values of initial velocities ξ0 and η0. Using this fact and operating
25
in a proper way we can eliminate the time from the two first equations ofEqs. (42) obtaining
ξ2
(η20 + ξ2
0)/k2
+η2
(η20 + ξ2
0k2)
= 1. (44)
Remark 6. Eq. (44) shows that the periodic orbits around collinear pointsare ellipses with centers located at these points. The parameters which de-termine these ellipses, i.e., the semi–major axis, the semi–minor axis, theeccentricity and the period of motion are a,b,e and T are respectively givenby
a =√
η20 + ξ2
0k2, b =
√η2
0 + ξ20k
2
k2,
e =
√1− 1
k2, T =
2π
τ.
Note that since η0 = −ξ0kτ if ξ0 6= 0 and ξ = 0 if η0 = 0, then the motionalong these orbits is retrograde.
7.2. Motion in the proximity of the triangular points.
7.2.1. Mean motion of the periodic orbits.
Since the characteristic polynomial of our system has four purely imagi-nary roots in neighborhood of the triangular points in the range 0 < µ < µc,the motion around these points is bounded and composed of two harmonicmotion. Thus, this motion in a synodic coordinate reference frame is givenby
ξ = C1 cos s1t + D1 sin s1t + C2 cos s2t + D2 sin s2t,η = C1 cos s1t + D1 sin s1t + C2 cos s2t + D2 sin s2t
(45)
where the terms with coefficients C1, D1, C1 and D1 are the long periodicterms while the rest correspond to the short periodic ones, see Abouelmagdet al. (2013) for more details. Moreover, s1 and s2 are the frequencies withrespect to long and short periodic orbits respectively given by
s21,2 =
1
2
[C ±
√D]. (46)
26
Thus, substituting Eqs. (37) and (38) into Eq. (46) using Eqs. (39) weobtain
s21 =
27
4µ
1 +23
4µ +
35
6A1 +
17
6B1 −
295
24A2
−115
24B2 +
241
24A2
1 −71
24B2
1 +307
12A1B1
,
s22 =
1− 27
4µ(1 +
23
4µ)− 3
2
(1 +
97
4µ
)A1 +
3
2
(1− 59
4µ
)B1
−45
8
(1 +
161
12µ
)A2 −
15
8
(1− 85
4µ
)B2
−2169
32µA2
1 +639
32µB2
1 −15
2
(1 +
2763
16µ
)A1B1
.
(47)
Remark 7. Eqs. (47) represents the frequencies of the long and short periodmotions when the effects of the oblateness are considered up to 10−6 of themain terms of potential and have been calculated to order µ.
7.2.2. Elliptic periodic orbits.
If we substitute Eqs. (45) into Eqs. (28) and equal the coefficients ofsine and cosine, the relations between the coefficients of the long and shortperiodic are given by
Ci = Γi
[2nsiDi − Ω0
xyCi
],
Di = −Γi
[2nsiCi − Ω0
xyDi
],
Γi =s2
i + Ω0xx
4n2s2i +
(Ω0
xy
)2 =1
s2i + Ω0
yy
(48)
where (i = 1, 2) while Ω0xx, Ω0
xy and Ω0yy are given by Eqs. (34).
If by ξ0, η0, ξ0 and η0 we denote the initial conditions at time t = 0, wenote that with a proper election of them either the short or the long periodicterms can be omitted from the solution. For instance if the short periodicones are cancel, will be held C2 = D2 = C2 = D2 = 0 and the relations
27
between the coefficients of Eqs. (45) and the initial conditions are given by
ξ0 = C1, η0 = C1,
ξ0 =Ω0
xyξ0 + η0(s21 + Ω0
yy)
2n, η0 = −
Ω0xyη0 + ξ0(s
21 + Ω0
xx)
2n,
D1 =Ω0
xyξ0 + η0(s21 + Ω0
yy)
2ns1
, D1 = −Ω0
xyη0 + ξ0(s21 + Ω0
xx)
2ns1
.
Now, let us consider the triangular points the origin of the coordinates sys-tem where the third body starts its motion from the origin of the coordinatesystem. Hence from Eqs. (26) and Eqs. (27) we have (ξ0, η0) = (−x0,−y0)where
ξ0 =1
2− µ +
1
2(A1 −B1) +
5
8(A2 −B2) +
5
8(A2
1 −B21) ,
η0 = ∓√
3
2
1− 1
3(A1 + B1) +
5
12(A2 + B2)
+1
36[7 (A2
1 + B21) + 68A1B1]
(49)
where the negative sign means that the infinitesimal body starts its motionfrom L4 while positive sign is linked to L5 .
Since the expansion of the potential function Ω around the triangularpoints L4,5 up to second order of (ξ, η) can be written as
Ω = Ω0 +1
2Ω0
xxξ2 + Ω0
xyξη +1
2Ω0
yyη2 (50)
where we have omitted the terms of order higher than two and Ω0 is givenby
Ω0 =
√3
2
1 +
5
6
(1− 2
5µ
)A1 +
1
2
(1 +
2
3µ
)B1 −
7
8
(1− 2
7µ
)A2
−5
8
(1 +
2
5µ
)B2 −
1
4µA2
1 −1
4(1− µ)B2
1
.
(51)Since
28
∣∣∣∣Ω0xx Ω0
xy
Ω0xy Ω0
yy
∣∣∣∣ =27
4µ(1− µ)
1 +
13
3(A1 + B1)−
20
3(A2 + B2)
+14
3(A2
1 + B21) +
95
6A1B1
> 0,
then the quadratic form of Eq. (50) shows that the trajectories of the in-finitesimal body around the triangular points are ellipses and periods of thesemotions are given by
Ti =2π
si
, i = 1, 2.
7.2.3. Orientation.
The quadratic expression of Ω in Eq. (50) contains the bilinear term ξηwhich appears as a result of the rotation of the principal axes of ellipses withrespect to the coordinates system (ξ, η) through an angle θ. Thus, we shallintroduce a new coordinates reference frame (ξ, η) called normal coordinatessuch that it makes disappear for the bilinear term and the orientation of theprincipal axes is given by
tan 2θ =2Ω0
xy
Ω0xx − Ω0
yy
, (52)
see Abouelmagd&El-Shaboury (2012) for more details on normal coordinates.
Remark 8. Substituting Eqs. (34) into Eq. (52) the orientations of theprincipal axes of the ellipses are controlled by
tan 2θ = ±√
3
1− 2µ +8
3(1− 2µ) A1 −
4
3(1− 2µ) B1
−10
3
(1− 5
4µ
)A2 +
5
3
(1− 1
2µ
)B2
+22
9
(1 +
241
44µ
)A2
1 +25
9
(1− 91
50µ
)B2
1
−70
9
(1− 106
35µ
)A1B1
where the positive sign furnishes the periodic motion around L4 while negativesign one around L5.
29
7.2.4. Semi–major and semi–minor axes.
Inspired in Abouelmagd et al. (2013), we can write Eq. (50) in normalcoordinates in the way
Ω = Ω0 +1
2λ1ξ
2 +1
2λ2η
2 (53)
where
λ1,2 =1
2
[Ω0
xx + Ω0yy ∓
√(Ω0
xx + Ω0yy
)2 − 4[Ω0
xxΩ0yy −
(Ω0
xy
)2]]and therefore
λ21 =
9
4µ
1− 1
4µ +
1
6(11A1 + 17B1)−
5
24(11A2 + 23B2)
− 1
24(223A2
1 + 71B21)−
31
4A1B1
,
λ22 = 3
1− 3
4µ(1− 1
4µ) +
5
2
(1− 19
20µ
)A1 +
3
2
(1− 3
4µ
)B1
−35
8µ(1− 27
28µ)A2 −
15
8
(1− 7
12µ
)B2
+1
32µ (223A2
1 + 71B21) +
5
2
(1 +
93
40µ
)A1B1
.
(54)
Since Eq.(50) and Eq. (53) allow the Jacobian integral c = 2Ω, thenequations of the ellipses are given by
ξ2
a2i
+η2
b2i
= 1
where ai and bi are respectively the lengths of the semi–major and semi–minor axes being
ai =
√ci − 2Ω0
λ1
,
bi =
√ci − 2Ω0
λ2
,
ci = 2Ω0 + Ω0xxξ
20 + 2Ω0
xyξ0η0 + Ω0yyη
20,
30
where (i = 1, 2) and the values of Ω0xx, Ω0
xy, Ω0yy; ξ0, η0; Ω0; λ1 and λ2 are
respectively given in Eqs. (34), (49),(51) and Eqs. (54). We note that Ω0xy
takes negative sign if the motion is around L4 when i = 1 and positive onearound L5 when i = 2.
7.2.5. Eccentricities of the ellipses.
The parameter which determines the eccentricities of the periodic orbits
is αi =1
2
(si + λ1/si
), (i = 1, 2), see Szebehely (1967) for more details. The
eccentricity value of the curves of zero velocity is controlled by α3 = λ1/λ2,see Abouelmagd et al. (2013). Consequently the values of the eccentricitiesfor all cases are given by
e2i = 1− α2
i ,
i = 1, 2, 3.
8. Conclusions
We study the existence of the libration points and their linear stabilityas well as periodic orbits around these points in the framework of restrictedthree–body problem when the primaries are oblate spheroids. The locationsof libration points are found and their stability is analyzed.
We observe that the triangular points are stable in the linear sense when0 < µ < µc and unstable for µc ≤ µ ≤ 1/2 where µc is the critical massratio which belongs to (0, 1/2) and depends on expressions that include thefactors of oblateness J2, J4 , J2 and J4. The collinear points are unstable.
We also prove that there is a periodic motion around the libration points.The trajectory of the infinitesimal body is represented by a ellipse. Therelations that give the values of the frequencies, the periods of periodic orbits,the lengths of semi–major and semi-minor axes, the eccentricities and theorientation of principal axes are stated. Furthermore the eccentricity valueof the curves of zero velocity is computed.
We emphasize that the results which we have obtained are different fromthe previous ones stated in the literature, because we consider the effect ofoblateness up to 10−6 of the main terms of the potential with respect to bothprimaries and this the first time which is studied.
This work generalize and extend the results from Abouelmagd (2012)where only was considered oblate spheroid the bigger primary, the results
31
from Abouelmagd&Sharaf (2013) where were consider the effects of twofirst zonal harmonic coefficients when the bigger body was radiating and thesmaller primary oblate and the results from Singh&Ishwar (1999) where wasconsidered the two primaries radiating and only were studied the effects upto J2.
It is worthy to underline that the model studied in the present paper isdirectly applicable to analyze planetary systems such as Earth–Moon, Mars–its Moons, Jupiter–its Moons or others.
Appendix
• At the point L1:
2sξ131 + [14s− 2sµ]ξ12
1 + [42s− 12sµ]ξ111
+
[1
4(−8 + 280s)− 30sµ
]ξ101
+
[1
4(−32 + 280s) +
1
4(−16− 160s) µ
]ξ91
+
[1
4(−32 + 168s− 12A1) +
1
4(−72− 120s + 12A1 − 12B1) µ
]ξ81
+
[1
4(−32 + 56s− 24A1) +
1
4(−128− 48s + 24A1 − 72B1) µ
]ξ71
+
1
4(−8 + 8s− 12A1 + 15A2)
+1
4µ (−112− 8s + 12A1 − 15A2 − 180B1 + 15B2)
ξ61
+1
4[−48− 240B1 + 90B2] µξ5
1 +1
4[−8− 180B1 + 225B2] µξ4
1
+1
4[−72B1 + 300B2] µξ3
1 +1
4[−12B1 + 225B2] µξ2
1
+45
2µξ1B2 +
15µB2
4= 0,
32
µ =
(8− 8s + 12A1 − 15A2) ξ6
1
15B2
−2(8 + 4s + 24A1 − 45A2)ξ71
15B2
+ O(ξ1)8
,
ξ1 =
(
15B2
8− 8s + 12A1 − 15A2
)1/6
µ1/6
+(8 + 4s + 24A1 − 45A2)
3√
45B2
[3(8− 8s + 12A1 − 15A2)]4/3
µ1/3
+ O(µ)1/2.
• At the point L2:
2sξ132 + [14s + 2sµ]ξ12
2 + [42s− 12sµ]ξ112
+
[1
4(8− 280s) +
1
4(−16 + 120s) µ
]ξ102
+
[1
4(−32 + 280s) +
1
4(80− 160s) µ
]ξ92
+
[1
4(48− 168s + 12A1) +
1
4(−168 + 120s− 12A1 − 12B1) µ
]ξ82
+
[1
4(−32 + 56s− 24A1) +
1
4(192− 48s + 24A1 + 72B1) µ
]ξ72
+
1
4(8− 8s + 12A1 − 15A2)
+1
4µ (−128 + 8s− 12A1 + 15A2 − 180B1 + 15B2)
ξ62
+1
4[48 + 240B1 − 90B2] µξ5
2 +1
4[−8− 180B1 + 225B2] µξ4
2
+1
4[72B1 − 300B2] µξ3
2 +1
4[−12B1 + 225B2] µξ2
2
−45
2µξ2B2 +
15µB2
4= 0,
33
µ =
(8− 8s− 12A1 + 15A2) ξ6
2
15B2
−2(8 + 4s + 24A1 − 45A2)ξ72
15B2
+ O(ξ2)8
,
ξ2 =
(
15B2
−8 + 8s− 12A1 + 15A2
)1/6
µ1/6
+(8 + 4s + 24A1 − 45A2)
3√
45B2
[3(8− 8s + 12A1 − 15A2)]4/3
µ1/3
+ O(µ)1/2.
• At the point L3:
34
2sξ133 + [12s + 2sµ]ξ12
3 + [30s + 12sµ]ξ113
+
[1
4(−8 + 160s) + 30sµ
]ξ103
+
[1
4(−48 + 120s) +
1
4(16 + 160s) µ
]ξ93
+
[1
4(−120 + 48s− 12A1) +
1
4(72 + 120s + 12A1 − 12B1) µ
]ξ83
+
[1
4(−160 + 8s− 72A1) +
1
4(128 + 48s + 72A1 − 24B1) µ
]ξ73
+
1
4µ (−120− 180A1 + 15A2)
+1
4(112 + 8s + 180A1 − 15A2 − 12B1 + 15B2)
ξ63
+
[1
4µ(48 + 240A1 − 90A2) +
1
4(−48− 240A1 + 90A2)
]ξ53
+
[1
4µ(8 + 180A1 − 225A2) +
1
4(−8− 180A1 + 225A2)
]ξ43
+
[1
4µ(72A1 − 300A2) +
1
4(−72A1 + 300A2)
]ξ33
+
[1
4µ(12A1 − 225A2) +
1
4(−12A1 + 225A2)
]ξ23
+45A2(1− µ)
2ξ3 +
15A2(1− µ)
4= 0,
35
µ =
1 +
(8s− 8− 12B1 + 15B2) ξ63
15A2
−2(8 + 4s + 24B1 − 45B2)ξ73
15A2
+ O(ξ3)8
,
ξ3 =
(
15A2
8− 8s + 12B1 − 15B2
)1/6
(1− µ)1/6
+(8 + 4s + 24B1 − 45B2)
3√
45A2
[3(8− 8s + 12B1 − 15B2)]4/3
(1− µ)1/3
+ O(1− µ)1/2,
where we recall that s = n2.
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR)at King Abdulaziz University, Jeddah, under grant no. (59 - 130 - 35 RG).The authors, therefore, acknowledge with thanks DSR technical and financialsupport. Also this work has been partially supported by MICINN/FEDERgrant number MTM2011–22587.
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