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Astrophys Space Sci (2013) 346:41–50DOI 10.1007/s10509-013-1420-7
O R I G I NA L A RT I C L E
The equilibrium points in the perturbed R3BP with triaxialand luminous primaries
Jagadish Singh
Received: 5 February 2013 / Accepted: 8 March 2013 / Published online: 9 April 2013© Springer Science+Business Media Dordrecht 2013
Abstract This study explores the effects of small perturba-tions in the Coriolis and centrifugal forces, radiation pres-sures and triaxiality of the two stars (primaries) on the posi-tion and stability of an infinitesimal mass (third body) in theframework of the planar circular restricted three-body prob-lem (R3BP). it is observed that the positions of the usualfive (three collinear and two triangular) equilibrium pointsare affected by the radiation, triaxiality and a small pertur-bation in the centrifugal force, but are unaffected by that ofthe Coriolis force. The collinear points are found to remainunstable, while the triangular points are seen to be stablefor 0 < μ < μc and unstable for μc ≤ μ ≤ 1
2 , where μc isthe critical mass ratio influenced by the small perturbationsin the Coriolis and centrifugal forces, radiation and triaxi-ality. It is also noticed that the former one and all the latterthree posses stabilizing and destabilizing behavior respec-tively. Therefore, the overall effect is that the size of the re-gion of stability decreases with increase in the values of theparameters involved.
Keywords Celestial mechanics · Perturbations · Radiationpressures · Triaxiality
1 Introduction
The restricted three-body problem (R3BP) in which twomassive bodies (primaries) revolve around their commoncentre of mass in circular orbits and a third body of neg-ligible mass moves in their gravitational field, is a simple
J. Singh (�)Department of Mathematics, Faculty of Science, Ahmadu BelloUniversity, Zaria, Nigeriae-mail: [email protected]
problem and has been receiving considerable attention ofscientists and astronomers because of its applications in dy-namics of the solar and stellar systems, lunar theory andartificial satellites. It possesses five points of equilibrium:three collinear and two triangular, where the gravitationaland centrifugal forces just balance each other. The collinearpoints are unstable, while the triangular points are stablefor the mass ratio μ < 0.03852 . . . (Szebehely 1967a). Theirstability occurs in spite of the fact that the potential energyhas a maximum rather than a minimum at the latter points.The stability is actually achieved through the influence ofthe Coriolis force, because the coordinate system is rotating(Wintner 1941; Contopoulos 2002).
By taking small perturbations in the Coriolis and cen-trifugal forces, various authors (Szebehely 1967b; Bhat-nagar and Hallan 1978; AbdulRaheem and Singh 2006;Singh 2009 and Singh and Begha 2011) have described theireffects on the motion of the third body. Szebehely (1967b)investigated the stability of triangular points by keeping thecentrifugal force constant and found that the Coriolis forceis a stabilizing force. Later, Subbarao and Sharma (1975)proved that this fact is not always true; as they observed anincrease in both the Coriolis and centrifugal forces due tooblateness of the primary. This was confirmed by AbdulRa-heem and Singh (2006).
The classical R3BP is not valid when at least one ofthe interacting bodies is an intense emitter of radiation.In certain stellar dynamics problems it is altogether inad-equate to consider solely the gravitational force. For ex-ample, when a star acts upon a particle in a cloud of gasand dust, the dominant factor is by no means gravity, butthe repulsive force of the radiation pressure (Radzievsky1950). In this connection, it is justifiable to modify themodel by superimposing a light repulsion field whose sourcecoincides with the source of the gravitational field of the
42 Astrophys Space Sci (2013) 346:41–50
main bodies. Several papers (Kunitsyn 2000; Singh 2009;Singh 2011) have discussed the effect of the radiation pres-sure in the R3BPs.
The bodies in the R3BP are strictly spherical in shape,but in nature, the celestial bodies are not perfect spheres.They are either oblate or triaxial. The Earth, Jupiter, Sat-urn, Regulus, Neutron stars and black dwarfs are oblate.The Moon, Pluto and its moon Charon are triaxial. Thelack of sphericity, triaxiality or oblateness of the celes-tial bodies causes large perturbations from a two-body or-bit. The motions of artificial Earth satellites are examplesof this. The most striking example of perturbations aris-ing from oblateness in the solar system is the orbit of thefifth satellite of Jupiter, Amalthea. This planet is so oblateand the satellite’s orbit is so small that its line of ap-sides advances about 900° in a year (Moulton 1914).Thisinspired several researchers (Subbarao and Sharma 1975;Elipe and Ferrer 1985; Sharma et al. 2001a, 2001b; Singh2011; Singh 2012 and Singh and Umar 2012a, 2012b) toinclude non sphericity of the bodies in their studies of theR3BP.
Our aim is to study the combined effects of small per-turbations in the Coriolis and centrifugal forces, triaxialityand radiation of the primaries on the position and linear sta-bility of the equilibrium points. This paper is organized asfollows: in Sect. 2, the equations governing the motion arepresented; Sect. 3 describes the positions of the equilibriumpoints, while their linear stability is analyzed in Sect. 4; thediscussion is given in Sect. 5, finally Sect. 6 summarizes theresults of this paper.
2 Equations of motion
Let m1 and m2 be the masses of the primary and secondarystars, respectively, and m be the mass of the third body. Us-ing dimensionless variables, the equations of motion of thethird body in a barycentric rotating coordinate system, canbe written as (Sharma et al. 2001a, 2001b; AbdulRaheemand Singh 2006):
x − 2αny = Ωx, y + 2αnx = Ωy, (1)
where
Ω = n2β
2
[(1 − μ)r2
1 + μr22
] + q1(1 − μ)
r1+ q2μ
r2
+ q1(1 − μ)
2r31
(2σ1 − σ2) − 3q1(1 − μ)
2r51
(σ1 − σ2)y2
+ 3q2μ(2σ ′1 − σ ′
2)
2r32
− 3μ(2σ ′1 − σ ′
2)y2q2
2r52
, (2)
n2 = 1 + 3
2(2σ1 − σ2) + 3
2
(2σ ′
1 − σ ′2
), (3)
r21 = (x − μ)2 + y2, r2
2 = (x + 1 − μ)2 + y2, (4)
μ = m2
m1 + m2≤ 1, m1 ≥ m2; σ1 = a2 − c2
5R2,
σ2 = b2 − c2
5R2, σ ′
1 = a′2 − c′2
5R2, σ ′
2 = b′2 − c′2
5R2,
σi, σ′i � 1, (i = 1,2).
Here μ is the mass parameter, n is the mean motionof the primaries; r1 and r2 are distances of the third bodyfrom the primaries. σ1, σ2 and σ ′
1, σ ′2 characterize the triax-
iality of the primary and secondary stars respectively witha, b, c and a′, b′, c′ as lengths of its semi-axes and R isthe dimensional distance between them. qi (i = 1,2) arethe radiation factors of the primary and secondary stars re-spectively and are given by Fpi
= Fgi(1 − qi) such that0 < (1 − qi) � 1 Radzievsky (1950), where Fgi and Fpi
arerespectively the gravitational and radiation pressure force.α = 1 + ε; |ε| � 1, β = 1 + ε′; |ε′| � 1 are parametersfor the Coriolis and centrifugal forces respectively to whichsmall perturbations ε and ε′ are given.
3 Location of equilibrium points
The equilibrium points represent stationary solutions of theR3BP. These points are the singularities of the manifold ofthe components of the velocity and the coordinates. There-fore, they can be found by setting x = 0 = y = x = y in theequations of motion.
3.1 Location of triangular points
The triangular points are the solutions of Ωx = 0, Ωy = 0with y �= 0; which yield
3x
[(1 − μ)q1(σ1 − σ2)
r51
+ μ(σ ′1 − σ ′
2)q2
r52
]
− 3(1 − μ)μ
[q1(σ1 − σ2)
r51
− (σ ′1 − σ ′
2)q2
r52
]
− (1 − μ)
[n2β − q1
r31
− 3
2
q1(4σ1 − 3σ2)
r51
+ 15
2
q1y2(σ1 − σ2)
r71
]= 0,
n2β − (1 − μ)q1
r31
− μq2
r32
− 3(1 − μ)
2r51
(4σ1 − 3σ2)q1
+ 15
2
(1 − μ)q1(σ1 − σ2)y2
r71
− 3
2
μ(4σ ′1 − 3σ ′
2)q2
r52
+ 15
2
μ(σ ′1 − σ ′
2)q2y2
r72
= 0.
(5)
When the primaries are neither radiating nor triaxial, ri =β−1/3 (i = 1,2). Therefore, when they are triaxial rigid bod-ies and sources of radiation, the values of ri will changeslightly, by εi , say, so that
ri = β−1/3 + εi, where |εi | � 1. (6)
Astrophys Space Sci (2013) 346:41–50 43
Fig. 1 Effect of radiation of theprimaries on the location oftriangular points (μ = 0.0365,
σ1 = 0.004, σ2 = 0.002, σ ′1 = 0.08,
σ ′2 = 0.02)
Substituting these values of ri into (4) and neglecting second
and higher order terms in εi , we obtain
x = μ − 1
2+ (ε2 − ε1)β
− 13 ,
y2 = β−2/3 − 1
4+ (ε1 + ε2)β
−1/3.
(7)
Putting the values of n2 from (3), ri from (6) and x, y from
(7) into (5) and restricting ourselves to only linear terms in
1 − qi , σi and σ ′i , we get
ε1 = β−1/3
3
[−(1 − q1) +
(15
8β4/3 − 3β2/3 − 3
)σ1
+(
−15
8β4/3 + 9
2β2/3 + 3
2
)σ2
+{
3
2
(μ
1 − μ
)β2/3 − 3
}σ ′
1
+{−3
2
(μ
1 − μ
)β2/3 − 3
}σ ′
2
],
ε2 = β−1/3
3
[−(1 − q2) +
(3β2/3
2μ− 3β2/3
2− 3
)σ1
+(
−3β2/3
2μ+ 3β2/3
2+ 3
2
)σ2
+(
15
8β4/3 − 3β2/3 − 3
)σ ′
1
+(
−15
8β4/3 + 9
2β2/3 + 3
2
)σ ′
2
].
(8)
The substitution of Eq. (8) in (7) results in
x = μ − 1
2+ β−2/3
[(1 − q1)
3− (1 − q2)
3
+ 1
2β2/3
(1
μ− 5
4β2/3 + 1
)σ1
+ 1
2β2/3
(− 1
μ+ 5
4β2/3 − 2
)σ2
+ 1
2β2/3
(5
4β2/3 − μ
1 − μ− 2
)σ ′
1
+ 1
2β2/3
(−5
4β2/3 + μ
1 − μ+ 3
)σ ′
2
],
y = ±√
4 − β2/3
2β1/3
[1 + 2
4 − β2/3
{−1
3(1 − q1) − 1
3(1 − q2)
+(
5
8β4/3 − 3
2β2/3 + β2/3
2μ− 2
)σ1
+(
−5β4/3
8+ 2β2/3 − β2/3
2μ+ 1
)σ2
+(
5β4/3
8+ μβ2/3
2(1 − μ)− β2/3 − 2
)σ ′
1
+(
−5β4/3
8+ 3β2/3
2− μβ2/3
2(1 − μ)+ 1
)σ ′
2
}].
(9)
Since r1 �= r2 �= 1, each of the two points represented by(9) form a scalene triangle with the primaries. These aredenoted by L4(x, y) and L5(x,−y) and are known as theLagrangian equilibrium points. One can see that the posi-tions of these points depend on the small perturbation in thecentrifugal force, triaxiality and radiation factors of the pri-maries. The absence of α show that they are not affected bya small change in the Coriolis force. Figure 1 shows that de-creasing radiation pressure factors q1 and q2 (0.99 & 9.95 to
44 Astrophys Space Sci (2013) 346:41–50
Fig. 2 Effect of triaxiality onthe location of the triangularpoints (μ = 0.0365,
q1 = 0.9997, q2 = 0.9998, β = 1.02,
σ1 = 0.004, σ2 = 0.002, σ ′2 = 0.02)
Fig. 3 Effect of a smallperturbation in centrifugal force(μ = 0.0365, q1 = 0.9998,
q2 = 0.9997, σ1 = 0.004,
σ2 = 0.002, σ ′1 = 0.08,
σ ′2 = 0.02)
0.65 & 0.6) brings the triangular points closer to the x-axis.The same effect is observed in Figs. 2 & 3 with increasingtriaxiality of the smaller primary σ ′
1 (0.0008 to 0.0045) anda small perturbation in centrifugal force β (1.0 to 1.02 andto 1.2) respectively.
3.2 Location of collinear points
The solutions of Ωx = 0, Ωy = 0, y = 0 yield the collinearpoints. That is,
f (x) = n2βx − (1 − μ)(x − μ)q1
r31
− μ(x − μ + 1)q2
r32
− 3(1 − μ)(x − μ)(2σ1 − σ2)q1
2r51
− 3μ(x − μ + 1)(2σ ′1 − σ ′
2)q2
2r52
= 0, (10)
where r1 = |x − μ|, r2 = |x + 1 − μ|.The abscissae of these points are the roots of (10). We
make use of a method based on the topological degree the-ory described in Kalantonis et al. (2001) to determine withcertainty the total number Nr of collinear points.
In order to avoid the situation whereby the denominatorof (10) would vanish at the points x = μ and x = μ − 1, we
Astrophys Space Sci (2013) 346:41–50 45
Table 1 Coordinates ofequilibrium points L1,2,3
Case xL1 xL2 xL3
1. −0.87668302 −1.07104368 0.74886500
2. −0.86350648 −1.09195321 1.01046690
3. −0.87766390 −1.07006281 0.74896424
4. −0.86353035 −1.09193160 1.01049230
5. −0.83711092 −1.10808256 0.75031535
6. −0.86435231 −1.09111106 1.01060499
7. −0.83781606 −1.10879919 0.75041039
8. −0.83782210 −1.10879171 0.75037480
9. −0.87670654 −1.07102196 0.74892937
10. −0.87670151 −1.07102762 0.74896250
11. −0.87668751 −1.07103849 0.74882956
12. −0.86433899 −1.09112292 1.01060326
13. −0.86434214 −1.09111793 1.01055514
apply the scheme (Picard 1892, 1922; Kavvadias and Vra-hatis 1996):
Nr = −γ
π
∫ b
a
f (x)f ′′(x) − f ′2(x)
f 2(x) + γ 2f ′2(x)dx
+ 1
πarctan
(γ [f (a)f ′(b) − f (b)f ′(a)]f (a)f (b) + γ 2f ′(a)f ′(b)
)(11)
γ is a small positive real constant), to the subintervals[a,μ − 1 − δ], [μ − 1 + δ,μ − δ] and [μ + δ, b] of the in-terval [a, b], where δ is a small real constant proportional tothe machine precision. We assume arbitrary values for μ, ε′,σi , σ ′
i (i = 1,2) and qi (i = 1,2) as 0.02545, 1.45 × 10−4,1.39×10−7, 1.32×10−4, 1.30×10−4, 1.28×10−4 and 0.4,0.5 respectively. We compute the integral of (11) and conse-quently obtain the total number of roots of (10) as three.
At each one of the aforementioned subintervals, we com-pute the coordinates of the collinear points with the use ofthe bisection method. Hence, we obtain
xL1 = −0.87668751, xL2 = −1.07103849,
xL3 = 0.74882956.
Using the same values for σi (i = 1,2), σ ′i (i = 1,2), μ, ε′
and qi (i = 1,2) as those of the present study, we obtain thecoordinates of the collinear equilibrium points (Table 1) fordifferent cases in order to show the effects of the various pa-rameters involved. The cases are classified in the followingorder:
Case 1. Primaries are radiating and triaxial rigid bodies;Case 2. Triaxiality of both primaries;Case 3. Both primaries are radiating;Case 4. Perturbations in the Coriolis and centrifugal forces
together with the triaxiality and oblateness of theprimary and secondary bodies respectively;
Case 5. Radiation pressure and oblateness of the primaryand secondary bodies respectively;
Case 6. Triaxiality of the primary body only;Case 7. Triaxiality and radiation pressure of the primary
body only;Case 8. Perturbations in the Coriolis and centrifugal forces
together with the triaxiality and radiation pressureof the primary body;
Case 9. Perturbations in the Coriolis and centrifugal forces,radiation pressure combined with the triaxialityand oblateness of the primary and secondary bod-ies respectively;
Case 10. Oblateness and radiation pressure of both pri-maries;
Case 11. The present problem (Perturbations in the Coriolisand centrifugal forces together with the triaxialityand radiation pressure of both primaries);
Case 12. The classical case (sphericity of both primaries);Case 13. Perturbations in the Coriolis and centrifugal forces
only.
4 Stability
We now examine the stability of an equilibrium configu-ration, that is, its ability to restrain the body motion in itsvicinity. To do so, we displace the third body a little from anequilibrium point with small velocity. If its motion is rapiddeparture from the vicinity of the point, we call such a po-sition of equilibrium an unstable one. If the body oscillatesabout the point, it is said to be a stable position.
Let the location of an equilibrium point be denoted by(x0, y0) and consider a small displacement (ξ, η) from thepoint such that x = x0 + ξ , and y = y0 + η. Substitutingthese values in (1), we obtain the variational equations
ξ − 2nαη = Ω0xxξ + Ω0
xyη,
η + 2nαξ = Ω0xyξ + Ω0
yyη.
46 Astrophys Space Sci (2013) 346:41–50
Their characteristic equation is
λ4 + (4n2α2 − Ω0
xx − Ω0yy
)λ2 + Ω0
xxΩ0yy − (
Ω0xy
)2 = 0,
(12)
where the superscript 0 indicates that the partial derivativesare evaluated at the equilibrium point (x0, y0).
4.1 Stability of triangular points
In the case of triangular points, we have
Ω0xx = 3(1 − μ)
(x0 − μ)2
r510
q1
× [1 + f + (σ1 − σ2)(x0 − μ)−2]
+ 3μ(x0 − μ + 1)2
r520
q2
× [1 + g + (
σ ′1 − σ ′
2
)(x0 − μ + 1)−2],
Ω0yy = 3(1 − μ)y2
0q1
r510
[1 + f + 10(σ1 − σ2)
r210
]
+ 3μy20q2
r520
[1 + g + 10(σ ′
1 − σ ′2)
r220
],
Ω0xy = 3(1 − μ)(x0 − μ)y0q1
r510
[1 + f + 5(σ1 − σ2)
r210
]
+ 3μ(x0 − μ + 1)y0
r520
q2
[1 + g + 5(σ ′
1 − σ ′2)
r220
],
f = 5
2
(2σ1 − σ2)
r210
− 35(σ1 − σ2)y20
2r410
= β2/3
8
[−100σ1 + 120σ2 + 35β2/3(σ1 − σ2)],
g = 5
2
(2σ ′1 − σ ′
2)
r220
− 35
2
(σ ′1 − σ ′
2)y20
r420
= β2/3
8
[−100σ ′1 + 120σ ′
2 + 35β2/3(σ ′1 − σ ′
2
)],
where ri0, x0 and y0 are given in Eqs. (6) to (9).Substituting these values into (12) and restricting our-
selves to linear terms in σi , σ ′i and 1 − qi , the characteristic
equation becomes
λ4 + bλ2 + c = 0 (13)
with
b = (4α2 − 3β
) + (12α2 − 9β
)σ1
+{−6α2 − 3β5/3(1 − μ) + 9β
2
}σ2
+ (12α2 − 9β
)σ ′
1 +{−6α2 − 3μβ5/3 + 9β
2
}σ ′
2,
c = 3μ(1 − μ)β8/3
2
× [3(4 − β2/3) + (
2 − β2/3){(1 − q1) + (1 − q2)}]
+ 9β8/3(1 − μ)
16
[16 − 36β2/3 + 10β4/3
+ μ(128 − 64β2/3 + 30β4/3 − 5β2)]σ1
+ 9β8/3(1 − μ)
16
[−16 + 36β2/3 − 10β4/3
+ μ(−64 + 52β2/3 − 30β4/3 + 5β2)]σ2
+ 9β8/3
16
[(144 − 100β2/3 + 40β4/3 − 5β2)μ
+ (−128 + 54β2/3 − 20β4/3 + 5β2)μ2]σ ′1
+ 9β8/3
16
[(−80 + 88β2/3 − 40β4/3 + 5β2)μ
+ (64 − 52β2/3 + 30β4/3 − 5β2)μ2]σ ′
2.
Its roots are
λ2 = −b ± √
2, (14)
where = b2 − 4c is the discriminant expressed as
= a1μ2 + a2μ + a3
with
a1 = β8/3[
9(4 − β2/3) + 6
(2 − β2/3){(1 − q1) + (1 − q2)
}
+ 9
4
(128 − 64β2/3 + 30β4/3 − 5β2)σ1
+ 9
4
(−64 + 52β2/3 − 30β4/3 + 5β2)σ2
+ 9
4
(128 − 44β2/3 + 20β4/3 − 5β2)σ ′
1
− 9
4
(64 − 52β2/3 + 30β4/3 − 5β2)σ ′
2
],
a2 = −9β8/3(4 − β2/3) − 6β8/3(2 − β2/3)
× {(1 − q1) + (1 − q2)
}
− 9β8/3
4
(112 − 28β2/3 + 20β4/3 − 5β2)σ1
+{
6β5/3(4α2 − 3β)
− 9β8/3
4
(−48 + 16β2/3 − 20β4/3 + 5β2)}σ2
− 9β8/3
4
(144 − 100β2/3 + 40β4/3 − 5β2)σ ′
1
+{−6β5/3(4α2 − 3β
)
− 9β8/3
4
(−80 + 88β2/3 + 5β2 − 40β4/3)}σ ′
2,
a3 = (4α2 − 3β
)2 +{
2(4α2 − 3β
)(12α2 − 9β
)
Astrophys Space Sci (2013) 346:41–50 47
− 9β8/3
4
(16 − 36β2/3 + 10β4/3)
}σ1
+{
2(4α2 − 3β
)(−6α2 + 9
2β − 3β5/3
)
− 9β8/3
4
(−16 + 36β2/3 − 10β4/3)}σ2
+ 2(4α2 − 3β
)(12α2 − 9β
)σ ′
1
+ 2(4α2 − 3β
)(−6α2 + 9
2β
)σ ′
2.
(15)
For simplicity, writing α = 1+ ε, |ε| � 1 and β = 1+ ε′,|ε′| � 1 in (15) and restrict ourselves to linear terms in ε, ε′,1 − qi , σi and σ ′
i , we obtain
a1 = 27
[1 + 22
9ε′ + 2(1 − q1)
9+ 2(1 − q2)
9+ 89
12σ1
− 37
12σ2 + 89
12σ ′
1 − 37
12σ ′
2
],
a2 = −27
[1 + 22
9ε′ + 2(1 − q1)
9+ 2(1 − q2)
9+ 33
4σ1
− 149
36σ2 + 79
12σ ′
1 − 73
36σ ′
2
],
a3 = 1 + 16ε − 6ε′ + 57
2σ1 − 63
2σ2 + 6σ ′
1 − 3σ ′2.
Now, we have
( )μ=0 = a3 > 0
and
( )μ= 1
2= 1
4a1 + 1
2a2 + a3 < 0.
Since ( )μ=0 and ( )μ= 1
2are of opposite signs, there is
only one value of μ, e.g., μC , in the open interval (0,1/2)
for which vanishes. This μC is called the critical massratio value. Therefore, we consider the three regions of thevalues of μ separately.
(i) when 0 ≤ μ < μc, > 0; the values of λ2 given by(14) are negative and therefore all the four characteris-tic roots are distinct pure imaginary numbers. Hence,the triangular points are stable.
(ii) When μC < μ ≤ 12 , < 0; the real parts of two of the
characteristic roots are positive. Therefore, the triangu-lar points are unstable.
(iii) When μ = μC , = 0; the values of λ2 given by(14) are same. This induces instability of the triangu-lar points.
Thus, for 0 ≤ μ < μc we have stability and for μC < μ ≤12 we have instability.
4.1.1 Critical mass
The solution of the quadratic equation = 0 for μ gives thecritical mass ratio value μC of the mass parameter. That is
μC = μ0 + μp + μr + μt (16)
where
μ0 = 1
2
(1 −
√23
27
),
μp = 4
27√
69
(36ε − 19ε′),
μr = − 2
27√
69
[(1 − q1) + (1 − q2)
],
μt = 1
2
(5
6+ 59
9√
69
)σ1 − 1
2
(19
18+ 85
9√
69
)σ2
+ 1
2
(−5
6+ 59
9√
69
)σ ′
1 + 1
2
(19
18− 85
9√
69
)σ ′
2.
Here, μ0 represents Routh’s critical mass ratio; while μp ,μr and μt are respectively the effects arising from small per-turbations in the Coriolis and centrifugal forces, radiationpressures, and triaxiality of the primaries. In the absenceof perturbations, μc reduces to the critical mass value ofthe R3BP when the primaries are triaxial rigid bodies andsource of radiations. This confirms the result of Sharma et al.(2001a). In this case μc < μ0, this shows that the size of theregion of stability decreases. Further in this case, if the pri-maries are non-luminous, μc becomes that of Sharma et al.(2001b). If there are no perturbations and the primaries areoblate radiating bodies (i.e., σ1 = σ2, σ ′
1 = σ ′2), the value of
μc agrees with that of Singh and Ishwar (1999) and Singhand Umar (2012b) when e = 0. But in the presence of per-turbations, this case verifies the results of AbdulRaheem andSingh (2006) that the Coriolis force has a stabilizing ten-dency, while the centrifugal force, radiation, and oblatenessof the primaries have destabilizing effects.
We now discuss the perturbing effects on the whole prob-lem.
For this, we consider the straight line AOB representedby the equation 36ε − 19ε′ = 0 in the (ε, ε′) plane (Fig. 4).This line divides the plane into two parts π1 and π2 as in-dicated. We call that part of the plane π1 which, standingat O and looking towards OA, is on our right. π2 is on theleft part. For the points in the part π1, 36ε − 19ε′ > 0 andμc < μ0. This implies that the size of the region of stabil-ity decreases. For the points in the part π2, 36ε − 19ε′ < 0and μc < μ0. This also implies that the size of the re-gion of stability decreases. For the point lying on the line36ε − 19ε′ = 0, μc < μ0. This again implies that the sizeof the region of stability decreases. For the points lying onthe ε-axis, ε′ = 0. That is, there is no perturbation in thecentrifugal force. Then
μc = μ0 + μr + μt + 16ε
3√
69,
48 Astrophys Space Sci (2013) 346:41–50
Fig. 4 The size of the region ofstability for 36ε − 19ε′ = 0 inthe (ε, ε′) plane
This shows that μc < μ0. If μr = 0, μt = 0 thenμc > μ0. Thus, keeping the centrifugal force constant inthe absence of triaxiality and radiation, the Coriolis forceremains a stabilizing force. But in the presence of either orboth of them, it is no longer a stabilizing force. This contra-dicts Szebehely’s (1967b) result, but confirms Subbarao andSharma’s (1975) result.
For the points lying on the ε′-axis, ε = 0. That is, there isno perturbation in the Coriolis force. Then
μc = μ0 + μr + μt − 76ε′
27√
69,
Here we see that μc < μ0. Hence, keeping the Coriolis forceconstant and in the absence or presence of either or boththe triaxiality and radiation pressures, the centrifugal forceis always destabilizing force. This case shows that triaxi-ality and radiation pressures make strong the destabilizingtendency of the centrifugal force. Further, we can observefrom Eq. (16) that the whole region of the mass parameter(0 ≤ μ ≤ 1
2 ) becomes unstable if the point (ε, ε′) lies on theline 4(36ε − 19ε′) + 27
√69(μ0 + μr + μt) = 0, because
μc = 0. And the whole region of the mass parameter exceptμ = 1
2 , (0 ≤ μ < 12 ) is stable if the point (ε, ε′) lies on the
line 4(36ε −19ε′)+27√
69(μ0 +μr +μt − 12 ) = 0 because
μ = 12 .
4.2 Stability of collinear points
First we examine the point lying in the interval (μ − 2,
μ − 1).For this point r1 = μ − x > 1, r2 = μ − 1 − x < 1, y = 0
Ωoxx = n2β + 2(1 − μ)q1
r31
+ 2μq2
r32
+ 6(1 − μ)q1(2σ1 − σ2)
r51
+ 6μq2(2σ ′1 − σ ′
2)
r52
Ω0xy = 0
Ω0yy = μ
[n2β
r1+ q2
r1r22
+ 3q2(2σ ′1 − σ ′
2)
2r1r42
− q2
r32
− 3(2σ ′1 − σ ′
2)
2r52
]− 3μ(σ ′
1 − σ ′2)
r52
− 3(σ1 − σ2)q1(1 − μ)
r51
Substituting β = 1 + ε′, |ε′| � 1, n2 = 1 + 32 (2σ1 − σ2) +
32 (2σ ′
1 − σ ′2), qi = [1 − (1 − qi)] with i = 1,2 and retaining
only linear terms in ε′, 1 − qi , σi and σ ′i , we get
Ωoxx = 1 + ε′ + 3
2(2σ1 − σ2) + 3
2
(2σ ′
1 − σ ′2
)
+ 2(1 − μ)[1 − (1 − q1)]r3
1
+ 2μ[1 − (1 − q2)]r3
2
+ 6(1 − μ)(2σ1 − σ2)
r51
+ 6μ(2σ ′1 − σ ′
2)
r52
> 0
Ω0xy = 0
Ω0yy = μ
[1
r1
(1 − 1
r32
)+ ε′
r1+ (1 − q2)
r1r32
+ 3(2σ1 − σ2)
2r1
+ 3(2σ ′1 − σ ′
2)
2r1
(1 − 1
r52
)]
− 3(σ1 − σ2)(1 − μ)
r51
− 3μ(σ ′1 − σ ′
2)
r52
< 0 (r2 < 1)
Similarly, for the points lying in (μ−1,0) and (μ,μ + 1),
Ω0xx > 0, Ω0
xy = 0 and Ω0yy < 0
Since Ω0xxΩ
0yy −(Ω0
xy)2 < 0, the discriminant of (12) is pos-
itive and the four roots of the characteristics equation (12)can be written as λ1 = b1, λ2 = −b1, λ3 = ib2 and λ4 =
Astrophys Space Sci (2013) 346:41–50 49
−ib2, where b1 and b2 are real. So, the solution is unstable.We may therefore conclude that the stability of the collinearpoints does not change due to perturbations in the Coriolisand centrifugal forces, triaxiality or radiation pressure forcesof the primaries and they remain unstable.
5 Discussion
Equations (1)–(4) describe the motion of a third body underthe influence of the gravitational field of stellar and triax-ial primaries together with small perturbations in the Cori-olis and centrifugal forces. Equations (9) and (10) give therespective positions of triangular and collinear equilibriumpoints, which are affected by the radiation pressures, triax-iality and the perturbation in the centrifugal force, but notby that of Coriolis force because Eqs. (9) and (10) are in-dependent of the parameter α. It may be noted here that inthis problem, the triangular points no longer form equilat-eral triangles with the primaries as they do in the classicalcase. Rather they form scalene triangles with the primaries.It is remarkable that the ordinates of the triangular pointsalso depend on the mass parameter μ. This is contrary to theclassical case, Singh and Ishwar (1999), AbdulRaheem andSingh (2006), and Singh and Umar (2012a, 2012b). Also,the second equation of (9) indicates that with a decrease inthe values of qi (i = 1,2), the triangular points move to-wards the line joining the primaries and their symmetricallocation with respect to this line is preserved.
Equation (16) gives the critical value of the mass param-eter μc of the system which depends upon small perturba-tions ε, ε′ in the Coriolis and centrifugal forces, triaxial-ity parameters σi , σ ′
i and radiation factors qi . This criticalvalue is used to determine the size of the region of stabil-ity of the triangular points and also helps in analyzing thebehavior of the parameters involved therein. It is obviousfrom (16) that the centrifugal force, radiation and triaxial-ity all have destabilizing effects, and therefore the size ofthe region of stability decreases with the increase of the val-ues of these parameters. Also, the presence of any one ormore of these parameters makes weak the stabilizing abil-ity of the Coriolis force. However, the net effect is that thesize of the region of the stability of the triangular points de-creases. In Sect. 4.2, all the aforementioned parameters areunable to change the instability character of the collinearpoints. In the absence of perturbations (i.e., ε = ε′ = 0), theresults obtained in this study are in agreement with thoseof Sharma et al. (2001b, 2001a) when the primaries arenon-luminous. When the primaries are oblate spheroids (i.e.,σ1 = σ2, σ ′
1 = σ ′2) and the perturbations are absent, the re-
sults coincide with those of Singh and Ishwar (1999), Singhand Umar (2012b) with circular orbits (i.e. e = 0). When theprimary is a non-luminous oblate body, and the secondary is
a luminous spherical body; the results tally with those ofSingh and Umar (2012a) in the case of circular orbits. Inthe presence of perturbations, when the primaries are oblatespheroids, the results correspond to those of AbdulRaheemand Singh (2006). When the primaries are non-luminous andthe secondary is oblate, the results agree with those of Singhand Begha (2011).
6 Conclusion
By considering both primaries as radiating and triaxial rigidbodies under the influence of small perturbations in the Cori-olis and centrifugal forces, we have determined the posi-tions of equilibrium points and have examined their linearstability. It is found that their positions are significantly af-fected by a small change in the centrifugal force, radiationand triaxiality factors of the primaries. It is further observedthat in spite of the introduction of aforesaid parameters, thecollinear points remain unstable. It is also seen that the tri-angular points are stable for 0 < μ < μc and unstable forμc ≤ μ ≤ 1/2, where μc is the critical mass parameter de-pendent on the small perturbations in the Coriolis and cen-trifugal forces, triaxiality and radiation factors; all of whichexcept the first (Coriolis force) have destabilizing tendenciesresulting in a decrease in the size of the region of stability.A practical application of this model could be the study ofthe motion of a dust grain particle near triaxial and luminousbinary systems.
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