robe’s problem: its extension to 2+2 bodies

Astrophys Space Sci (2012) 339:283–294DOI 10.1007/s10509-012-0991-z

O R I G I NA L A RT I C L E

Robe’s problem: its extension to 2 + 2 bodies

Bhavneet Kaur · Rajiv Aggarwal

Received: 2 October 2011 / Accepted: 10 January 2012 / Published online: 27 January 2012© Springer Science+Business Media B.V. 2012

Abstract In the problem of 2 + 2 bodies in the Robe’ssetup, one of the primaries of mass m∗

1 is a rigid sphericalshell filled with a homogeneous incompressible fluid of den-sity ρ1. The second primary is a mass point m2 outside theshell. The third and the fourth bodies (of mass m3 and m4

respectively) are small solid spheres of density ρ3 and ρ4 re-spectively inside the shell, with the assumption that the massand the radius of third and fourth body are infinitesimal. Weassume m2 is describing a circle around m∗

1. The masses m3

and m4 mutually attract each other, do not influence the mo-tion of m∗

1 and m2 but are influenced by them. We also as-sume masses m3 and m4 are moving in the plane of motionof mass m2. In the paper, the equations of motion, equilib-rium solutions, linear stability of m3 and m4 are analyzed.There are four collinear equilibrium solutions for the givensystem. The collinear equilibrium solutions are unstable forall values of the mass parameters μ,μ3,μ4. There exist aninfinite number of non collinear equilibrium solutions eachfor m3 and m4, lying on circles of radii λ,λ′ respectively(if the densities of m3 and m4 are different) and the centreat the second primary. These solutions are also unstable forall values of the parameters μ,μ3,μ4, ϕ, ϕ′. Such a modelmay be useful to study the motion of submarines due to theattraction of earth and moon.

Keywords Robe’s restricted problem · Equilibriumsolution · Stability · Buoyancy force

B. Kaur (�)Lady Shri Ram College for Women, University of Delhi, Delhi,Indiae-mail: [email protected]

R. AggarwalSri Aurobindo College, University of Delhi, Delhi, India

1 Introduction

Robe (1977) has considered a new kind of restricted three-body problem in which one of the primaries of mass m∗

1 is arigid spherical shell filled with a homogeneous incompress-ible fluid of density ρ1. The second primary is a mass pointm2 outside the shell. The third body of mass m3 is a smallsolid sphere of density ρ3 supposed moving inside the shell,subjected to the attraction of m2 and the buoyancy force dueto the fluid ρ1. He assume that the mass and radius of thethird body are infinitesimal. He has shown the existence ofan equilibrium solution with mass m3 at the centre of theshell while the mass m2 describes a Keplerian orbit aroundmass m∗

1. Further, he has discussed the linear stability of thisequilibrium solution. He has explicitly discussed two cases.In the first case, the orbit of m2 around m∗

1 is circular and inthe second case, the orbit is elliptic, but the shell is empty(i.e. no fluid inside it) or the densities of m∗

1 and m3 areequal. In each case, the domain of stability has been inves-tigated for the whole range of parameters occurring in theproblem.

In the above problem, the existence of only one equi-librium solution namely, centre of the first primary is dis-cussed. Hallan and Rana (2001a) studied the existence andthe linear stability of all the equilibrium solutions in theRobe’s restricted three-body problem. They (2001b) stud-ied the location and stability of the equilibrium points in theRobe’s circular restricted three body problem when smallperturbations are given to the coriolis and centrifugal forces.They further (2003) generalised the above problem with thedensity parameter having arbitrary value. Again they (2004)studied the effect of oblateness on the location and stabilityof equilibrium points in Robe’s circular problem.

Hallan and Mangang (2007) studied the existence andlinear stability of equilibrium points in the Robe’s restricted

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284 Astrophys Space Sci (2012) 339:283–294

three body problem when the first primary is an oblatespheroid. They (2008) studied the effect of perturbations incoriolis and centrifugal forces on the nonlinear stability ofequilibrium point in Robe’s restricted circular three bodyproblem.

Whipple and Szebehely (1984) generalized the originalform of circular restricted three body problem to n ≥ 2 pri-mary bodies. The number of bodies v of small mass not per-turbing the primaries was increased from v = 1 to v ≥ 1.The minor bodies did not affect the motion of the primaries.They discussed the regions of motion and applications ofthis problem.

Whipple (1984) studied equilibrium solutions of the re-stricted problem of 2 + 2 bodies. He further studied the lin-ear stability of all the equilibrium solutions.

Srivastava and Garain (1991) studied the effect of smallperturbation in the coriolis and centrifugal forces on the lo-cation of the equilibrium point in the Robe’s problem bytaking the orbit of m2 around m1 as a circle and assumingdensities of m1 and m3 to be equal.

Plastino and Platino (1995) considered the Robe’s prob-lem by taking the shape of the fluid body as Roche’s ellip-soid (Chandrashekhar 1987). They discussed the linear sta-bility of the equilibrium solution, which is the centre of theellipsoid.

Giordano et al. (1997) discussed the effect of drag forceon the stability of the equilibrium point, both in the Robe’sproblem (1977) and the problem studied by Plastino andPlatino (1995). They discussed that in the first case, fourregions of stability out of five are changed to instability,whereas in the second case regions of stability remained un-changed.

In this paper, we shall study the case of the restrictedproblem of 2 + 2 bodies in the Robe’s setup. Such a modelmay be useful to study the motion of submarines due to theattraction of earth and moon.

2 Statement of the problem and equations of motion

In the problem of 2 + 2 bodies in the Robe’s setup, one ofthe primaries of mass m∗

1 is a rigid spherical shell filled withhomogeneous incompressible fluid of density ρ1. The sec-ond primary is a mass point m2 outside the shell. The thirdand the fourth bodies (of mass m3 and m4 respectively) aresmall solid spheres of density ρ3 and ρ4 respectively insidethe shell, with the assumption that the mass and the radiusof the third and the fourth body are infinitesimal. We assumem2 is describing a circle around m∗

1 with constant angularvelocity ω (say). The masses m3 and m4 mutually attracteach other, do not influence the motion of m∗

1 and m2 butare influenced by them. We also assume masses m3 and m4

are moving in the plane of motion of mass m2.

Fig. 1 Geometry of the Robe’s restricted problem of 2 + 2 bodies

As in the case of classical restricted problem (Szebehely1967), let the orbital plane of m2 around m∗

1 (i.e. shell withits fluid of density ρ1) be taken as the ξη plane and let theorigin of the coordinate system be at the centre of mass O

of the two finite bodies. The coordinate system Oξη are asshown in the Fig. 1. Various forces acting on m3 are:

1. The gravitational force F32 acting on m3 due to m2 is

F32 = Gm3m2R32

R332

.

2. The gravitational force F34 acting on m3 due to m4 is

F34 = Gm3m4R34

R334

.

3. The gravitational force FA exerted by the fluid of densityρ1 on m3 is

FA = −(

4

3

)πGρ1m3R13,

where Rij = MiMj, M1 is the centre of the shell m∗1 and

M3 the centre of m3.4. The buoyancy force FB acting on m3 is

FB =(

4

3

)π

Gρ21m3R13

ρ3.

The last expression derived is taking into account that m3

is a sphere of very small radius b, so that the pressure ofthe fluid ρ1 inside the shell keeps its spherical symmetryaround M1. The buoyancy force is then ( 4

3πb3)ρ1g, whereg is the gravity of the fluid ρ1 at M3 i.e.

g =(

4

3

)πGρ1R13

with

m3 =(

4

3πb3

)ρ3.

Astrophys Space Sci (2012) 339:283–294 285

The equation of motion of m3 in the inertial system is

R̈ = Gm2R32

R332

+ Gm4R34

R334

− 4

3πGρ1

(1 − ρ1

ρ3

)R13,

where R = OM3 and Rij = MiMj.Now, we determine the equation of motion of m3 in the

synodic system. Let us suppose that the coordinate sys-tem Oξη rotates with angular velocity ω. This is the sameas the angular velocity of m2 which is describing a circlearound m∗

1.In the rotating (synodic) system, the equation of motion

of m3 is

∂2r∂t2

+ 2ω × ∂r∂t

+ ω × (ω × r)

= Gm2R32

R332

+ Gm4R34

R334

− 4

3πGρ1

(1 − ρ1

ρ3

)R13 (1)

where r = OM3 and ω = ωk̂ = (constant).Let the coordinates of m3 and m4 be (ξ, η) and (ξ ′, η′)

respectively.The equations of motion of m3 in cartesian coordinates

are

ξ̈ − 2ωη̇ = − Gm2(ξ − x2)

[(ξ − x2)2 + η2] 32

− Gm4(ξ − ξ ′)[(ξ − ξ ′)2 + (η − η′)2] 3

2

− 4

3πGρ1

(1 − ρ1

ρ3

)(ξ − x1) + ω2ξ, (2)

η̈ + 2ωξ̇ = − Gm2η

[(ξ − x2)2 + η2] 32

− Gm4(η − η′)[(ξ − ξ ′)2 + (η − η′)2] 3

2

− 4

3πGρ1(1 − ρ1

ρ3)η + ω2η. (3)

Now, as m2 is moving around m∗1 in a circle of radius ‘a’

with angular velocity ω, we have

ω =√

G(m∗1 + m2)

a3.

We, now, fix the units such that m∗1 + m2 = 1, a = 1. We

choose t in such a way that G = 1.We further take

μ1 = m∗1

m∗1 + m2

, μ2 = m2

m∗1 + m2

so that μ1 + μ2 = 1.

Let μ2 = μ, (say), then μ1 = 1 − μ. Coordinates of m∗1

and m2 are (−μ,0), (1 − μ,0).In the new units ω = 1.The equations of motion of m3 in the dimensionless

cartesian coordinates are

ξ̈ − 2η̇ = Vξ , (4)

and

η̈ + 2ξ̇ = Vη, (5)

where

V = 1

2

(ξ2 + η2

)+ μ

R32+ μ4

R34− K

2

[(ξ + μ)2 + η2

](6)

and

μ4 = m4

m∗1 + m2

� 1, K = 4

3πρ1

(1 − ρ1

ρ3

). (7)

Similarly, in the rotating (synodic) system, the equations ofmotion of m4 in dimensionless cartesian coordinates are

ξ̈ ′ − 2η̇′ = V ′ξ ′ , (8)

η̈′ + 2ξ̇ ′ = V ′η′ , (9)

where

V ′ = 1

2

(ξ ′2 + η′2) + μ

R42+ μ3

R43− K ′

2

[(ξ ′ + μ

)2 + η′2]

(10)

and

μ3 = m3

m∗1 + m2

� 1, K ′ = 4

3πρ1

(1 − ρ1

ρ4

). (11)

3 Equilibrium solutions

The equilibrium solutions of m3 and m4 are given by

Vξ = 0 = Vη; V ′ξ ′ = 0 = V ′

η′

i.e.,

ξ − μ{ξ − (1 − μ)}

R332

− μ4(ξ − ξ ′)

R334

− K(ξ + μ) = 0, (12)

η − μη

R332

− μ4(η − η′)

R334

− Kη = 0 (13)

and

ξ ′ − μ{ξ ′ − (1 − μ)}

R342

− μ3(ξ ′ − ξ)

R343

− K ′(ξ ′ + μ) = 0,

(14)


η′ − μη′

R342

− μ3(η′ − η)

R343

− K ′η′ = 0 (15)

Case I: Collinear Equilibrium Solutions

By inspection, we see that (13) and (15) are satisfied withη = η′ = 0.

It remains to determine ξ and ξ ′ such that the followingsimplified forms of (12) and (14) are satisfied, i.e.

ξ − μ{ξ − (1 − μ)}|ξ − (1 − μ)|3 − μ4

(ξ − ξ ′)|ξ − ξ ′|3 − K(ξ + μ) = 0, (16)

ξ ′ − μ{ξ ′ − (1 − μ)}|ξ ′ − (1 − μ)|3 − μ3

(ξ ′ − ξ)

|ξ ′ − ξ |3 − K ′(ξ ′ + μ) = 0.

(17)

When μ4 = 0, there exists only one equilibrium solution ofthe system, viz (−μ,0), (Hallan and Rana 2001a). Again,when μ3 = 0, (−μ,0) is the only equilibrium solution ofthe system. We now apply the perturbation theory when nei-ther μ3 nor μ4 is zero. We further define

(x,y) = 1

2(x2 + y2) + μ

[{x − (1 − μ)}2 + y2] 12

− K

2[(x + μ)2 + y2]

and

′(x, y) = 1

2(x2 + y2) + μ

[{x − (1 − μ)}2 + y2] 12

− K ′

2[(x + μ)2 + y2].

The solutions ξ and ξ ′ of (16) and (17) may be expressed aspower series in small parameters

εi = μi

(�μ3 + μ4)23

� 1, (i = 3,4) (18)

where

� = (3)xx

′(4)xx

.

The upper suffix (3) and (4) denote the evaluation of thederivatives at the equilibrium solution (−μ,0) for m3 andm4 respectively. We take

ξ = −μ +∞∑

j=1

a1j εj

4 , (19)

ξ ′ = −μ +∞∑

j=1

a2j εj

3 (Whipple and Szebehely 1984).

(20)

The equations (12), (13), (14), and (15) can be written as

x(ξ, η) − μ4(ξ − ξ ′)

R334

= 0, (21)

y(ξ, η) − μ4(η − η′)

R334

= 0, (22)

′x(ξ

′, η′) − μ3(ξ ′ − ξ)

R343

= 0, (23)

′y(ξ

′, η′) − μ3(η′ − η)

R343

= 0. (24)

To o(ε), where ε = max(ε3, ε4), by Taylor’s series, we have

x(ξ,0) = a11ε4(3)xx ,

′x(ξ

′,0) = a21ε3′(4)xx .

The equations (21) and (23) when η = η′ = 0 become

a11ε4(3)xx − μ4

(ξ − ξ ′)|ξ − ξ ′|3 = 0, (25)

a21ε3′(4)xx − μ3

(ξ ′ − ξ)

|ξ ′ − ξ |3 = 0. (26)

The equations (25) and (26) may be combined to yield

μ3a11ε4(3)xx + μ4a21ε3

′(4)xx = 0.

Since μ3ε4 = μ4ε3, we have

a11(3)xx + a21

′(4)xx = 0 or

a21 = −�a11.

(27)

To o(ε), using (19), (20), and (27) in (25), we have

|a11|3 = μ4

ε4(3)xx (ε4 + �ε3)2

.

Now, we have from (18)

ε4 + �ε3 = (μ4 + �μ3)13 .

Therefore,

a11 = ± 1

((3)xx )

13

.

Also,

a21 = −�a11 = ± 1

((3)xx )

13

(3)xx

′(4)xx

.


Therefore, to o(ε), from (19), we have

ξ = −μ + a11ε4

= −μ ± μ4

[(μ4 + �μ3)2(3)xx ] 1

3

.

and from (20), we have

ξ ′ = −μ + a21ε3

= −μ ± 1

((3)xx )

13

(3)xx

′(4)xx

μ3

(μ4 + �μ3)23

.

Also,

(3)xx = 1 − K + 2μ,

′(4)xx = 1 − K ′ + 2μ.

Therefore,

ξ = −μ ± μ4

[(μ4 + �μ3)2(1 − K + 2μ)] 13

(28)

and

ξ ′ = −μ ± (1 − K + 2μ)23

(1 − K ′ + 2μ)

μ3

(μ4 + �μ3)23

. (29)

In the case when ρ3 = ρ4, we have

I. K = K ′II. (x,y) = ′(x, y)

III. (3)xx = 1 − K + 2μ = 1 − K ′ + 2μ =

′(4)xx

IV. � = 1

Therefore, ξ and ξ ′ become

ξ = −μ ± μ4

[(μ3 + μ4)2(1 − K + 2μ)] 13

, (30)

and

ξ ′ = −μ ± μ3

[(μ3 + μ4)2(1 − K + 2μ)] 13

. (31)

Thus, (ξ,0) and (ξ ′,0) give approximate locations of thecollinear equilibrium solutions of the Robe’s restrictedproblem of 2 + 2 bodies. There are two possible posi-tions of the equilibrium solution each for m3 and m4.Let us denote them by ξ31, ξ32 corresponding to m3 andξ ′

41, ξ ′42 corresponding to m4. Therefore, the positions of

collinear equilibrium solutions of m3 and m4 are given by(ξ31,0), (ξ32,0) and (ξ ′

41,0), (ξ ′42,0) respectively. The po-

sitions of the collinear equilibrium solutions are illustratedin Fig. 2.

Case II: Non Collinear Equilibrium Solutions

In this case, we assume η �= 0, η′ �= 0. The equilibrium so-lutions of m3 and m4 are given by (12), (13), (14), and (15).

Fig. 2 Location of collinear equilibrium solutions of the Robe’s re-stricted problem of 2 + 2 bodies. Circles denote the positions of m3and triangles denote the positions of m4

The solutions of these equations may be expressed as powerseries in small parameters ε3 and ε4 such that

ξ = x′ +∞∑

j=1

a1j εj

4 , (32)

η = y′ +∞∑

j=1

b1j εj

4 , (33)

ξ ′ = x′′ +∞∑

j=1

a2j εj

3 , (34)

η′ = y′′ +∞∑

j=1

b2j εj

3 , (35)

where

εi = μi

(�μ3 + μ4)23

� 1 (i = 3,4) (36)

and

� = (3)xx

′(4)xx

.

The upper suffix (3) and (4) denote the evaluation of thederivatives at the equilibrium solution (x′, y′), (x′′, y′′) form3 and m4 respectively. Here (x′, y′) is any point lying onthe circle

{(1 − μ − x)2 + y2} =(

μ

1 − K

) 23 = λ2 (say)

therefore, x′ = 1 − μ − λ cos(φ), y′ = λ sin(φ) (Hallan andRana 2001a).

Also, (x′′, y′′) is any point lying on the circle

{(1 − μ − x)2 + y2} =(

μ

1 − K ′

) 23 = λ

′2 (say),

therefore, x′′ = 1 − μ − λ′ cos(φ′), y′′ = λ′ sin(φ′), 0 ≤φ, φ′ ≤ 2π .


The equations (12), (13), (14), and (15) can be written as(21), (22), (23), and (24).

To o(ε), where ε = max(ε3, ε4) and using the values ofξ, η, ξ ′, η′ from (32), (33), (34) and (35) and applyingTaylor’s series, the first term of (21) is

x(ξ, η) = a11ε4(3)xx + b11ε4

(3)xy . (37)

Similarly, we can determine y(ξ, η), ′x(ξ

′, η′), and′

y(ξ′, η′).

Since μ3,μ4, ε3, ε4 are very small, ignoring the higherorder terms and using (32), (33), (34) and (35), the secondterm of (21) is

μ4(ξ − ξ ′)

R334

= μ4(x′ − x′′)

[(x′ − x′′)2 + (y′ − y′′)2] 32

. (38)

Similarly

μ4(η − η′)

R334

= μ4(y′ − y′′)

[(x′ − x′′)2 + (y′ − y′′)2] 32

. (39)

Using (37), (38), (39), the equations (21), (22), (23), (24)become(a11

(3)xx + b11

(3)xy

)ε4 + μ4t1 = 0, (40)

(a11

(3)xy + b11

(3)yy

)ε4 + μ4t2 = 0, (41)

(a21

′(4)xx + b21

′(4)xy

)ε3 − μ3t1 = 0, (42)

(a21

′(4)xy + b21

′(4)yy

)ε3 − μ3t2 = 0, (43)

where

t1 = − (x′ − x′′)[(x′ − x′′)2 + (y′ − y′′)2] 3

2

, (44)

t2 = − (y′ − y′′)[(x′ − x′′)2 + (y′ − y′′)2] 3

2

. (45)

From (40) and (41), we have

a11 = μ4((3)xy t2 −

(3)yy t1)

ε4((3)xx

(3)yy − (

(3)xy )2)

,

b11 = μ4((3)xy t1 −

(3)xx t2)

ε4((3)xx

(3)yy − (

(3)xy )2)

.

From (42) and (43), we have

a21 = μ3(′(4)yy t1 −

′(4)xy t2)

ε3(′(4)xx

′(4)yy − (

′(4)xy )2)

,

b21 = μ3(′(4)xx t2 −

′(4)xy t1)

ε3(′(4)xx

′(4)yy − (

′(4)xy )2)

.

To o(ε), we get

ξ = x′ + a11ε4

= x′ + μ4((3)xy t2 −

(3)yy t1)

((3)xx

(3)yy − (

(3)xy )2)

= X1 (say), (46)

η = y′ + b11ε4

= y′ + μ4((3)xy t1 −

(3)xx t2)

((3)xx

(3)yy − (

(3)xy )2)

= Y1 (say). (47)

Also,

ξ ′ = x′′ + a21ε3

= x′′ + μ3(′(4)yy t1 −

′(4)xy t2)

(′(4)xx

′(4)yy − (

′(4)xy )2)

= X2 (say), (48)

η′ = y′′ + b21ε3

= y′′ + μ3(′(4)xx t2 −

′(4)xy t1)

(′(4)xx

′(4)yy − (

′(4)xy )2)

= Y2 (say), (49)

where

(3)xx = 3μ(Cos(φ))2

λ3,

(3)xy = −3μCos(φ)Sin(φ)

λ3,

(3)yy = 1 − μ

λ3+ 3μ(Sin(φ))2

λ3,

′(4)xx = 3μ(Cos(φ′))2

λ′3 ,

′(4)xy = −3μCos(φ′)Sin(φ′)

λ′3 ,

′(4)yy = 1 − μ

λ′3 + 3μ(Sin(φ′))2

λ′3

and t1, t2 are given by (44), (45).Equations (46), (47), (48) and (49) give approximate lo-

cations of the non-collinear equilibrium solutions of theRobe’s restricted problem of 2 + 2 bodies. The positionsof the non-collinear equilibrium solutions are illustrated inFig. 3. It may be noted that there are an infinite number ofnon-collinear equilibrium solutions as φ, φ′ lie between 0and 2π .

In the case when ρ3 = ρ4, we have

I. K = K ′II. (x,y) = ′(x, y)

Therefore, to o(ε)

ξ = x′ + a11ε4


Fig. 3 Location of non collinear equilibrium solutions (ρ3 �= ρ4) ofthe Robe’s restricted problem of 2 + 2 bodies

= x′ + μ4((3)xy t2 −

(3)yy t1)

((3)xx

(3)yy − (

(3)xy )2)

. (50)

η = y′ + b11ε4

= y′ + μ4((3)xy t1 −

(3)xx t2)

((3)xx

(3)yy − (

(3)xy )2)

. (51)

Also,

ξ ′ = x′′ + a21ε3

= x′′ + μ3((4)yy t1 −

(4)xy t2)

((4)xx

(4)yy − (

(4)xy )2)

, (52)

η′ = y′′ + b21ε3

= y′′ + μ3((4)xx t2 −

(4)xy t1)

((4)xx

(4)yy − (

(4)xy )2)

. (53)

where

(3)xx = 3μ(Cos(φ))2

λ3,

(3)xy = −3μCos(φ)Sin(φ)

λ3,

(3)yy = 1 − μ

λ3+ 3μ(Sin(φ))2

λ3,

(4)xx = 3μ(Cos(φ′))2

λ3,

(4)xy = −3μCos(φ′)Sin(φ′)

λ3,

(4)yy = 1 − μ

λ3+ 3μ(Sin(φ′))2

λ3

and t1, t2 are given by (44), (45).In this particular case, (50), (51), (52) and (53) give ap-

proximate locations of the non-collinear equilibrium solu-

Fig. 4 Location of non collinear equilibrium solutions (ρ3 = ρ4) ofthe Robe’s restricted problem of 2 + 2 bodies

tions of the Robe’s restricted problem of 2 + 2 bodies. Thepositions of the non-collinear equilibrium solutions are il-lustrated in Fig. 4.

4 Applications

We consider the two primaries Earth (m∗1) and Moon (m2)

and the infinitesimal masses, two submarines m3 and m4.The following is the physical data of the submarines.

1. Class and Type: Shishumar class submarine (m3).Displacement: 1427 long tons surfaced.

: 1821 long tons submerged.m3 = 1427 long tons = 1449832 kg (1 long ton =

1016 kg)ρ3 = 804.79 kg/m3.

2. Class and Type: Sindhughosh class submarine (m4).Displacement: 2288 long tons surfaced.

: 3027 long tons submerged.m4 = 2288 long tons = 2324608 kg (1 long ton =

1016 kg)ρ4 = 776.27 kg/m3.

Density of salt water (ρ1) = 1027 kg/m3.

Units of mass:

• In dimensionless units, m∗1 + m2 = 1.

• Mass of Earth (m∗1) = 5.9742 × 1024 kg.

• Mass of Moon (m2) = 7.3483 × 1022 kg.• Thus, 1 kg = 1

6.048×1024 new unit.

Therefore,

μ2 = m2

m∗1 + m2

∼= 0.01,

μ3 = m3

m∗1 + m2

∼= 10−18,

μ4 = m4

m∗1 + m2

∼= 10−17.


Table 1 Collinear equilibrium solutions for m3 and m4 (ρ3 �= ρ4)

μ ξ31 ξ32 ξ ′41 ξ ′

42

0.0100 −0.00999992 −0.01000008 −0.009999990 −0.01000010

Units of distance:

Distance between Earth and Moon = 384400 km = 1 newunit.

Thus, 1 metre = 1384400000 new unit.

Hence,

ρ1 = 9317.7 new units,

ρ3 = 7301.65 new units,

ρ4 = 7040.45 new units,

K = 4

3πρ1

(1 − ρ1

ρ3

)= −10780.8,

K ′ = 4

3πρ1

(1 − ρ1

ρ4

)= −12629.4.

Table 1 gives the collinear equilibrium solutions for m3 andm4 using the above data. The numerical solutions are correctin all digits printed.

Tables 2 and 3 give non-collinear equilibrium solutionsfor m3 and m4 respectively with μ = 0.01 and φ′ = φ + 30◦(supposed) using the above data. The numerical solutionsare correct in all digits printed.

5 Stability of the equilibrium solutions

Let (X,Y ) be one of the equilibrium solution. Let it bedisplaced to (X + α,Y + β) where α,β � 1. Then ξ =X + α,η = Y + β .

The equations of motion (4) and (5) become

α̈ − 2β̇ = Vξ (X + α,Y + β) ,

β̈ + 2α̇ = Vη (X + α,Y + β) .

It maybe noted that for m3 it is V and for m4 it is V ′.Therefore, by Taylor’s series, we have,

α̈ − 2β̇ = αV(0)ξξ + βV

(0)ξη , (54)

β̈ + 2α̇ = αV(0)ξη + βV (0)

ηη (55)

where the upper suffix (0) denotes the evaluation of thederivatives at the equilibrium solution (X,Y ).

The characteristic equations of m3 and m4 are given by

λ41 − λ2

1

(V

(3)ξξ + V (3)

ηη − 4)

+(

V(3)ξξ V (3)

ηη −(V

(3)ξη

)2)

= 0. (56)

λ42 − λ2

2

(V

′(4)

ξ ′ξ ′ + V′(4)

η′η′ − 4)

+(

V′(4)

ξ ′ξ ′ V′(4)

η′η′ −(V

′(4)

ξ ′η′)2

)

= 0. (57)

Here the upper suffix (3) and (4) denote the evaluation ofthe derivatives at the equilibrium solutions.

The equation (56) has two roots in λ21. Let us denote them

by �11 and �12. Similarly the two roots of (57) in λ22 are

denoted by �21 and �22. The four roots of (56) in λ1 willbe ±√

�11,±√�12. Similarly the four roots of (57) in λ2

will be ±√�21,±√

�22.We have

λ21 = (V

(3)ξξ + V

(3)ηη − 4) ± √

d

2(58)

where

d =(V

(3)ξξ + V (3)

ηη − 4)2 − 4

(V

(3)ξξ V (3)

ηη −(V

(3)ξη

)2)

and

λ22 = (V

′(4)

ξ ′ξ ′ + V′(4)

η′η′ − 4) ± √d1

2(59)

where

d1 =(V

′(4)

ξ ′ξ ′ + V′(4)

η′η′ − 4)2 − 4

(V

′(4)

ξ ′ξ ′ V′(4)

η′η′ −(V

′(4)

ξ ′η′)2

).

In the case when ρ3 = ρ4 we have K = K ′. The eigenvaluescan be deduced from (58) and (59) by replacing K ′ by K .

If any of the roots of the characteristic equations (56) and(57) is positive and real, the corresponding equilibrium so-lution will be unstable.

Stability of collinear equilibrium solutions The positionsof collinear equilibrium solutions of m3 and m4 are givenby (ξ31,0), (ξ32,0) and (ξ ′

41,0), (ξ ′42,0) respectively. Four

cases arise.

Case I. The equilibrium solutions for m3 and m4 are at(ξ31,0), (ξ ′

41,0). Equation (56) has four roots in λ1. Letus denote them by ±λ11,±λ12. Similarly the roots of (57)in λ2 be denoted by ±λ21,±λ22.


Tabl

e2

Non

colli

near

equi

libri

umso

lutio

nsfo

rm

3w

ithμ

=0.

01an

dφ

′ =φ

+30

◦(ρ

3�=

ρ4)

φx

′y

′ξ

η

00.

9802

4780

3504

3664

6614

2983

00

0.98

0247

8035

0436

6461

6853

682

3.55

4323

7278

8550

9843

5657

00×

10−1

7

300.

9815

5435

0092

0838

0492

3959

20.

0048

7609

8247

8167

4514

0381

656

0.98

1554

3500

9208

3815

5744

760

0.00

4876

0982

4781

6786

1821

7687

7

600.

9851

2390

1752

1832

2863

0599

40.

0084

4564

9907

9161

8098

2703

288

0.98

5123

9017

5218

3301

8027

670

0.00

8445

6499

0791

6252

0691

7784

6

900.

9899

9999

9999

9999

9111

8215

80.

0097

5219

6495

6334

9028

0763

311

Com

plex

Infin

ityIn

dete

rmin

ate

120

0.99

4876

0982

4781

6753

6058

322

0.00

8445

6499

0791

6180

9827

0328

80.

9948

7609

8247

8168

4460

8458

70.

0084

4564

9907

9161

0989

6228

731

150

0.99

8445

6499

0791

6177

3124

724

0.00

4876

0982

4781

6745

1403

8165

60.

9984

4564

9907

9161

9825

7409

50.

0048

7609

8247

8167

0409

8586

434

180

0.99

9752

1964

9563

3516

0934

486

00.

9997

5219

6495

6335

2055

1063

4−3

.554

3237

2788

5509

8435

6570

0×

10−1

7

210

0.99

8445

6499

0791

6177

3124

724

−0.0

0487

6098

2478

1674

5140

3816

560.

9984

4564

9907

9161

6666

1955

6−0

.004

8760

9824

7816

7861

8217

6877

240

0.99

4876

0982

4781

6753

6058

322

−0.0

0844

5649

9079

1618

0982

7032

880.

9948

7609

8247

8166

8043

3664

7−0

.008

4456

4990

7916

2520

6917

7846

270

0.98

9999

9999

9999

9991

1182

158

−0.0

0975

2196

4956

3349

0280

7633

11C

ompl

exIn

finity

Inde

term

inat

e

300

0.98

5123

9017

5218

3228

6305

994

−0.0

0844

5649

9079

1618

0982

7032

880.

9851

2390

1752

1831

3762

7972

9−0

.008

4456

4990

7916

1098

9622

8731

330

0.98

1554

3500

9208

3804

9239

592

−0.0

0487

6098

2478

1674

5140

3816

560.

9815

5435

0092

0837

8397

9022

1−0

.004

8760

9824

7816

7040

9858

6434

360

0.98

0247

8035

0436

6466

1429

830

00.

9802

4780

3504

3664

6168

5368

23.

5543

2372

7885

5098

4356

5700

×10

−17


Tabl

e3

Non

colli

near

equi

libri

umso

lutio

nsfo

rm

4w

ithμ

=0.

01an

dφ

′ =φ

+30

◦(ρ

3�=

ρ4)

φx

′′y

′′ξ

′η

′

00.

9819

8829

3360

1630

2260

5036

10.

0046

2556

0985

1781

7159

5488

816

0.98

1988

2933

6016

3020

9801

195

0.00

4625

5609

8517

8134

6633

5668

8

300.

9853

7443

9014

8218

0130

8130

50.

0080

1170

6639

8369

4376

5045

188

0.98

5374

4390

1482

1793

5692

132

0.00

8011

7066

3983

6879

7967

1591

0

600.

9899

9999

9999

9999

9111

8215

80.

0092

5112

1970

3563

4319

0977

633

Com

plex

Infin

ityIn

dete

rmin

ate

900.

9946

2556

0985

1781

8092

8301

10.

0080

1170

6639

8369

4376

5045

188

0.99

4625

5609

8517

8173

8937

807

0.00

8011

7066

3983

7007

7333

7446

7

120

0.99

8011

7066

3983

6959

6313

955

0.00

4625

5609

8517

8171

5954

8881

60.

9980

1170

6639

8369

5841

3162

70.

0046

2556

0985

1782

0852

7620

945

150

0.99

9251

1219

7035

6370

7383

864

00.

9992

5112

1970

3563

7091

4485

63.

1984

1646

3919

3752

0663

2102

×10

−17

180

0.99

8011

7066

3983

6959

6313

955

−0.0

0462

5560

9851

7817

1595

4888

160.

9980

1170

6639

8369

6125

6312

1−0

.004

6255

6098

5178

1346

6335

6688

210

0.99

4625

5609

8517

8180

9283

011

−0.0

0801

1706

6398

3694

3765

0451

880.

9946

2556

0985

1781

8866

7218

4−0

.008

0117

0663

9836

8797

9671

5910

240

0.98

9999

9999

9999

9991

1182

158

−0.0

0925

1121

9703

5634

3190

9776

33C

ompl

exIn

finity

Inde

term

inat

e

270

0.98

5374

4390

1482

1801

3081

305

−0.0

0801

1706

6398

3694

3765

0451

880.

9853

7443

9014

8218

0834

2650

9−0

.008

0117

0663

9837

0077

3337

4467

300

0.98

1988

2933

6016

3022

6050

361

−0.0

0462

5560

9851

7817

1595

4888

160.

9819

8829

3360

1630

2382

3268

9−0

.004

6255

6098

5178

2085

2762

0945

330

0.98

0748

8780

2964

3611

4980

452

00.

9807

4887

8029

6436

1132

1946

0−3

.198

4164

6391

9375

2066

3210

2×

10−1

7

360

0.98

1988

2933

6016

3022

6050

361

0.00

4625

5609

8517

8171

5954

8881

60.

9819

8829

3360

1630

2098

0119

50.

0046

2556

0985

1781

3466

3356

688


Table 4 Stability of collinear equilibrium solutions for m3 (ρ3 �= ρ4)

μ ±λ11 ±λ12 ±λ13 ±λ14

0.01 ±209.7086117726478780465229 ±76.27799490163028638359341i ±175.0584407696850689792168 ±29.12181592076459177443333

μ ±λ15 ±λ16 ±λ17 ±λ18

0.01 ±175.0584407696534164331535 ±29.12181592085973434748463 ±209.7086117726214566887602 ±76.27799490159396574617318i

Table 5 Stability of non-collinear equilibrium solutions for m3 with μ = 0.01 and φ′ = φ + 30◦(ρ3 �= ρ4)

φ (degree) ±√�11 ±√

�12

0 ±179.8374382023723853403734 ±0.00001608049389644347143414969

30 ±179.8374382023725398901880 ±0.00001513658262928219503520821

60 ±Indeterminate ±Indeterminate


120 ±179.8374382023729966183127 ±0.00001229815020923473960583077

150 ±179.8374382023733063735463 ±9.574681259814230138424997 × 10−6

180 ±179.8374382023723853403734 ±0.00001608049389644347143414969

210 ±179.8374382023725398901880 ±0.00001513658262928219503520821



300 ±179.8374382023729966183127 ±0.00001229815020923473960583077

330 ±179.8374382023733063735463 ±9.574681259814230138424997 × 10−6

360 ±179.8374382023723853403734 ±0.00001608049389644347143414969

Case II. The equilibrium solutions for m3 and m4 are at(ξ31,0), (ξ ′

42,0).

Equation (56) has four roots in λ1. Let us denote them by±λ13,±λ14. Similarly the roots of (57) in λ2 be denoted by±λ23,±λ24.

Case III. The equilibrium solutions for m3 and m4 are at(ξ32,0), (ξ ′

41,0). Equation (56) has four roots in λ1. Let usdenote them by ±λ15,±λ16. Similarly the roots of (57) inλ2 be denoted by ±λ25,±λ26.

Case IV. The equilibrium solutions for m3 and m4 are at(ξ32,0), (ξ ′

42,0). Equation (56) has four roots in λ1. Let usdenote them by ±λ17,±λ18. Similarly the roots of (57) inλ2 be denoted by ±λ27,±λ28.

If any of the four roots described above in each case arepositive and real, the equilibrium solution is unstable other-wise stable.

Table 4 gives the four roots of (56) in λ1 for each of thecases described above for μ = 0.01 with K �= K ′. Since oneof the roots of the characteristic equation (56) is always pos-itive and real,the equilibrium solution is unstable. Similarobservation is seen in the case when K = K ′. In a similaranalysis for m4, we observe that all the equilibrium solu-tions are unstable.

Stability of non collinear equilibrium solutions Table 5gives the four roots of equation (56) corresponding to μ =0.01 and φ(0 ≤ φ ≤ 360◦) with φ′ = φ +30◦ (supposed)andK �= K ′. Since one of the roots of the characteristic equationis always positive and real, the equilibrium solution is unsta-ble. Similar observation is seen in the case when K = K ′. Ina similar analysis for m4, we observe that all the equilibriumsolutions are unstable.

6 Discussion

In the Robe’s restricted three body problem,the point(−μ,0), the centre of the first primary, is an equilibrium so-lution for all values of the density parameter K , mass param-eter μ, eccentricity parameter e. Hallan and Rana (2001a)proved that besides (−μ,0), other equilibrium solutions ex-ist only when K �= 0 and when the second primary movesaround the first in a circular orbit. In addition to the existingcollinear equilibrium solution, they showed the existenceof one more which is stable only under certain restrictionson K and μ. There are two triangular equilibrium solutionswhich are always unstable. In addition, there are infinitenumber of equilibrium solutions lying on a circle. The cir-cular equilibrium solutions are always unstable.


Fig. 5 φ against thediscriminant d for μ = 0.01

In our problem, there are four collinear equilibrium solu-tions for the given system namely (ξ31,0), (ξ ′

41,0); (ξ31,0),(ξ ′

42,0); (ξ32,0), (ξ ′41,0); (ξ32,0), (ξ ′

42,0) (Fig. 2). Thecollinear equilibrium solutions are unstable for the pa-rameters μ,μ3,μ4. There exist an infinite number of noncollinear equilibrium solutions each for m3 and m4, lyingon circles of radii λ, λ′ respectively (if the densities of m3

and m4 are different) and the centre at the second primary(Fig. 3). These solutions are also unstable for the parametersμ, μ3, μ4, φ, φ′.

For μ = m2m∗

1+m2= 0.01, Fig. 5 plots φ against the dis-

criminant of (58). We observe that the discriminant is neverzero whatever φ maybe. The discontinuity in the graph ap-pears at those values of φ at which the discriminant is inde-terminate.

For μ = 0, the non collinear equilibrium solutions ofRobe’s restricted problem of 2 + 2 bodies are topologi-cally equivalent to the collinear equilibrium solutions andare therefore unstable.

The introduction of the parameter μ′ = μ5−i

μ3+μ4permits

analysis of the stability of mi for i = 1,2 and for a giventotal mass of the minor bodies. Without loss of generality,only the stability of m3 will be considered. When μ′ = 0, theproblem is reduced to the Robe’s restricted problem of threebodies consisting of m3,m

∗1,m2. For μ′ ∼= 1, m3 can be re-

garded as infinitesimally small mass while m4 is a small, butfinite, mass perturbing the motion of m3.

References

Chandrashekhar, S.: Ellipsoidal Figures of Equilibrium. Dover, NewYork (1987), Chap. 8

Giordano, C.M., Plastino, A.R., Platino, A.: Robe’s restricted threebody problem with drag. Celest. Mech. Dyn. Astron. 66, 229–242(1997)

Hallan, P.P., Rana, N.: The existence and stability of equilibriumpoints in the Robe’s restricted problem three-body problem. Ce-lest. Mech. Dyn. Astron. 79, 145–155 (2001a)

Hallan, P.P., Rana, N.: Effect of perturbation in coriolis and centrifugalforces on the location and stability of the equilibrium point in theRobe’s circular restricted three body problem. Planet. Space Sci.49, 957–960 (2001b)

Hallan, P.P., Rana, N.: Effect of perturbation in the coriolis and cen-trifugal forces on the location and stability of the equilibriumpoints in the Robe’s circular problem with density parameter hav-ing arbitrary value. Indian J. Pure Appl. Math. 34(7), 1045–1059(2003)

Hallan, P.P., Rana, N.: Effect of oblateness on the location and stabilityof equilibrium points in Robe’s circular problem. Indian J. PureAppl. Math. 35(3), 40–43 (2004)

Hallan, P.P., Mangang, K.B.: Existence and linear stability of equilib-rium points in the Robe’s restricted three body problem when thefirst primary is an oblate spheroid. Planet. Space Sci. 55, 512–516(2007)

Hallan, P.P., Mangang, K.B.: Effect of perturbations in coriolis and cen-trifugal forces on the nonlinear stability of equilibrium point inRobe’s restricted circular three body problem. Adv. Astron. 55,512–516 (2008)

Plastino, A.R., Platino, A.: Robe’s restricted three body problem revis-ited. Celest. Mech. Dyn. Astron. 61, 197–206 (1995)

Robe, H.A.G.: A new kind of three body problem. Celest. Mech. Dyn.Astron. 16, 243–351 (1977)

Srivastava, A.K., Garain, D.: Effect of perturbation on the location oflibration point in the Robe restricted problem of three bodies. Ce-lest. Mech. Dyn. Astron. 51, 67–73 (1991)

Szebehely, V.S.: Theory of Orbits. Academic Press, New York (1967)Whipple, A.L., Szebehely, V.S.: The restricted problem of n+v bodies.

Celest. Mech. 32, 137 (1984)Whipple, A.L.: Equilibrium solutions of the restricted problem of 2+2

bodies. Celest. Mech. 33, 271–294 (1984)

robe’s problem: its extension to 2+2 bodies

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