^{1}

^{*}

^{2}

The present work studies the stability condition of central control artificial equilibrium points of the planar circular restricted problem of 2 + 2 bodies (PCRP2 + 2B) and also its variant when the shape of larger mass is taken to be an oblate spheroid. We find that the paper will be of great application in choosing an artificial equilibrium point (AEP) in the neighbourhood of numerous planets e.g. Jupiter or the bodies which provide a model of the problem studied. The minimum thrust will save a quantum of energy to be applied to have an arbitrary point as a chosen starter. For solar sailing and magnetic force this minimum thrust will be of great use.

The 2 + 2 body problem has been introduced by [

The problem under our investigation is really a generalization of the classical restricted problem of three bodies (CRP3B). In nature we find a system of primary masses interacting with each other and their motion is completely determined by their interactions. On the other hand they are not influenced by the presence of the minor bodies. For example, we may consider the motion of asteroids or comets in the gravitational field of the Jupiter and Sun or that lunar probe in the gravitational field of Earth and Moon. In these problems we find that the presence of the minor bodies does not affect the motion of primaries while the primaries affect the latter ones. The (RP2 + 2B) has been investigated by many others as referred above.

Here we shall investigate the presence of AEP with the application of propellant force exerted by solar sail or other mechanical devices for the (CRP2 + 2B). The use of solar sails is not a new idea, but it is being exercised since the last many years. The names of [

In the present paper we have studied the existence of (AEP) for the (RP2 + 2B). We foresee a great prospect of achievement with the present study. The stationing of the space-crafts in the stable region of the libration point with the minimum thrust is our main aim behind the present study. It is proposed that two space-crafts may be stationed in the neighborhood of stable libration points with Earth &Moon or with Jupiter and Sun or similarly other planetary bodies for the primaries. The result can be utilized to study the motion of minor planets or comets.

Here we have chosen the model of (PRP2 + 2B) as introduced by [

In the present work we consider the motion of two minor bodies with negligible mass interacting with each other but not affecting the two primaries interacting with each other and also affecting the motion of minor bodies. While studying the motion when one of the larger mass is of the shape of an oblate spheroid, we have used the result [_{1} may be taken independent of the motion of m_{2} and similarly the motion of m_{2} may be taken independent of m_{1}, where m_{1} and m_{2} are infinitesimally small. This idea of independence splits the problem into two independent restricted problems and the whole analysis reduces to the study made in [

Here the problem of motion of two minor bodies with masses m_{1} and m_{2} is being investigated under the gravitational field of two primary bodies with masses M_{1} and M_{2} (M_{1} ≥ M_{2}) moving along circular Keplerien orbits about their centre of mass where m_{1} and m_{2} being ≤M_{2}. The orbital plane is the common plane [

Referred to the above synodic system when the origin is taken at the smaller primary, the differential equation describing the restricted problem of three bodies may be written as:

x ¨ i − 2 y ˙ = 1 μ i ∂ T ∂ x i y ¨ i + 2 x ˙ = 1 μ i ∂ T s ∂ y i (2.1)

where, (.) denotes the differential coefficient with respect to t and

T = ∑ i = 1 2 μ i [ 1 2 ( 1 − μ + x i ) 2 + y i 2 ] + 1 − μ δ 1 i + μ ρ 2 i + 1 2 μ 3 − i R + a x i x i + a y i y i , μ = M 2 M 1 + M 2 , μ i = m i M 1 + M 2 , δ 1 i 2 = ( x i + 1 ) 2 + y i 2 , ρ 2 i 2 = x i 2 + y i 2 , R 2 = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 , l = M 1 M 2 = Dimensional length = 1 ( chosen as dimensionless ) .

It has been examined [

The control acceleration for maintaining a desired equilibrium point (x_{i0}, y_{i0}) will be given as:

a x i 0 = ( 1 − μ ) 1 + x i 0 δ ( 1 i ) 0 3 + μ x i 0 ρ ( 2 i ) 0 3 − x i 0 + μ − 1 − μ 3 − i ∂ ∂ x i ( 1 R ) , a y i 0 = ( 1 − μ ) y i 0 δ ( 1 i ) 0 3 + μ y i 0 ρ ( 2 i ) 0 3 − y i 0 − μ 3 − i ∂ ∂ y i ( 1 R ) . ( i = 1 , 2 )

In order to consider the linear stability condition we shall prefer to write the Hamiltonian with respect to the origin taken to be the equilibrium point (x_{i}_{0}, y_{i}_{0}), so that

ξ i = x i − x i 0 , η i = y i − y i 0 , p ξ i = p x i − p x i 0 , p η i = p y i − p y i 0 . ( i = 1 , 2 )

where ξ i , η i , p ξ i & p η i are the variable in the canonic variables x i , y i , p x i & p y i as introduced in [

1 2 ∑ i = 1 2 [ ( p ξ i 2 + p η i 2 ) + p ξ i η i − p ξ i η i − p η i ξ i + ξ i ( μ − 1 − x i 0 ) − η i y i 0 − 1 − μ δ ( 1 i ) 0 − μ ρ ( 2 i ) 0 − 1 2 μ 3 − i R − a x i ξ i − a y i η i ]

δ ( 1 i ) 0 = ( 1 + ξ i + x i 0 ) 2 + ( η i + y i 0 ) 2 , ρ ( 2 i ) 0 = ( ξ i + x i 0 ) 2 + ( η i + y i 0 ) 2 .

It has been seen [_{1} is independent of the motion of the minor body with mass µ_{2} thus it is seen that the problem reduces to two restricted problem with µ_{1} and µ_{2} respectively. Taking the advantage of this conclusion our problem reduces to that studied by [

H = H 0 + H 2 ( μ 1 ) + H 2 ( μ 2 ) ,

where

H 2 ( μ 1 ) = 1 2 ( p ξ 1 2 + p η 1 2 ) + n ( p ξ 1 η 1 − ξ 1 p η 1 ) + a 1 ξ 1 2 + b 1 ξ 1 η 1 + c 1 η 1 2 ,

H 2 ( μ 2 ) = 1 2 ( p ξ 2 2 + p η 2 2 ) + n ( p ξ 2 η 2 − ξ 2 p η 2 ) + a 2 ξ 2 2 + b 2 ξ 2 η 2 + c 2 η 2 2 _{ }

a i , b i , & c i , may be expressed as [

Thus the linearized equation of motion may be written as

p ˙ ξ i = − ∂ H 2 ∂ ξ i = p η i − 2 a i ξ i − b i η i ,

p ˙ η i = − ∂ H 2 ∂ η i = p ξ i − 2 c i η i − b i ξ i ,

ξ ˙ i = ∂ H 2 ∂ p ξ i = p ξ i + n η i (3.1)

η ˙ i = ∂ H 2 ∂ p η i = p η i − n ξ i

The corresponding characteristic equation may be written as

λ 4 + 2 f i λ 2 + g i = 0 , ( i = 1 , 2 ) . (3.2)

where f i = a i + c i + 1 = 2 δ 1 i 3 − 1 2 δ 1 i 3 + μ ρ 2 i 3 − δ 1 i 3 2 ρ 2 i 3 δ 1 i 3

g i = − b i 2 + 2 a i ( 2 c i − 1 ) − 2 c i + 1 = δ 1 i 6 − 2 + δ 1 i 3 δ 1 i 6 + μ 4 ρ 2 i 5 δ 1 i 6 ( − 4 ρ 2 i 5 δ 1 i 3 + 16 ρ 2 i 5 + 4 ρ 2 i 2 δ 1 i 6 + 18 δ 1 i 3 − 9 δ 1 i 5 + 2 ρ 2 i 2 δ 1 i 3 − 9 δ 1 i ρ 2 i 4 − 9 δ 1 i ) − μ 2 4 ρ 2 i 6 δ 1 i 6 ( − 9 ρ 2 i δ 1 i 5 + 18 ρ 2 i δ 1 i 3 − 9 ρ 2 i 5 δ 1 i + 18 ρ 2 i 2 δ 1 i − 8 δ 1 i 6 − 9 ρ 2 i δ 1 i + 8 ρ 2 i 6 + 2 δ 1 i 3 ρ 2 i 3 ) , ( i = 1 , 2 ) (3.3)

In order to have the linear stability the following conditions are to be satisfied

a) f i 2 − g i > 0 , b) f i > 0 , ( i = 1 , 2 ) c) g i > 0. (3.4)

With the existence of stable region in the referred problem our next step will be to identify the stable point inside the region for which the required control acceleration is as small as possible.

In this regard our problem will be to fix the equilibrium distance ρ_{(}_{2i)0} from the second primary and to search the space craft position that provides minimum control acceleration. For this we shall minimize the objective function.

J i = a x i 2 + a y i 2 ( i = 1 , 2 ) (4.1)

J i = μ ( ρ 2 i 3 − 1 ) ( ρ 2 i − 1 ) ( μ + ρ 2 i 3 + ρ 2 i ) ρ 2 i 4 − 1 − μ ρ 2 i 3 δ 1 i 4 × [ ( μ − ρ 2 i 3 ) δ 1 i 6 − ( μ − 2 ρ 2 i 3 + μ ρ 2 i 3 ) δ 1 i 3 + μ ( ρ 2 i 3 − 1 ) ( ρ 2 i 2 − 1 ) δ 1 i − ( 1 − μ ) ρ 2 i 3 ] .

which is written just by putting an index i with δ_{1} & δ_{2} in corresponding expressions in the referred work.

For minimum value of J i , we shall put

∂ J i ∂ δ 11 = 0 = ∂ J 2 ∂ δ 12

Where the corresponding expression as given in the referred paper may be written which we shall not write for the sake of repetition. Solving δ_{11} & δ_{12} by the perturbation method and restricting to µ^{2}, we have

δ i ( o p t ) = δ 1 i ( 0 ) + μ δ 1 i ( 1 ) + μ 2 δ i ( 2 ) = 1 + 1 − ρ 2 i 3 6 ρ 2 i μ − ρ 2 i 8 + 2 ρ 2 i 6 − 2 ρ 2 i 5 + 2 ρ 2 i 3 + ρ 2 i 2 − 4 36 ρ 2 i 4 , ( i = 1 , 2 )

The corresponding control acceleration restricting to 0 (µ^{2}) will be written as

a i = a x i 2 + a y i 2 = μ ρ 2 i 1 − ρ 2 i 2 + 8 ρ 2 i 3 − 2 ρ 2 i 5 − 4 ρ 2 i 6 + ρ 2 i 8 4 , ( i = 1 , 2 )

and the corresponding dimensional acceleration may be written as

a i = G M 2 ρ 2 i 2 l 2 1 − ρ 2 i 2 + 8 ρ 2 i 3 − 2 ρ 2 i 5 − 4 ρ 2 i 6 + ρ 2 i 8 4

For an approximate analytical estimation of the distance from the second primary allowing the stability may be written as

ρ 2 i 6 − 26 ρ 2 i 3 μ + ρ μ 2 ≻ 0 , ρ 2 i 3 − μ ≻ 0 , − 2 ρ 2 i 8 + 3 ρ 2 i 6 − ρ 2 i 5 + 8 ρ 2 i 3 − 16 μ ≻ 0.

From the first two conditions we have

ρ 2 i > ρ 2 i ( min ) = ( 13 µ + 4 µ 10 ) 1 3 ≈ ( 26 µ ) 1 3 ^{ }

and

a ( ρ 2 i ( min ) ) = ( 26 μ ) 1 3 | 1 − 26 μ | 4 − ( 26 μ ) 2 3 52

Finally, the minimum control acceleration will be

r min ≃ l ( 26 μ ) 1 3 , and a ^ ( r min ) ≃ G M 2 μ l 2 [ ( 26 μ ) 1 3 | 1 − 26 μ | 4 − ( 26 μ ) 2 3 52 ]

Since the expression for r min & â ( r min ) are independent of the indices (i), so they will be the expressions for both minor bodies.

Here we shall use the notations of the work [

x ¨ − 2 n y ˙ i = 1 μ i ∂ Γ ∂ x i , y ¨ + 2 n x ˙ i = 1 μ i ∂ Γ ∂ y i . (5.1)

where

Γ = ∑ i = 1 2 μ i [ 1 2 n { ( x i + 1 − μ ) 2 + y i 2 } + 1 − μ δ 1 i + μ ρ 2 i + ( 1 − μ ) 2 δ 1 i 3 + 1 2 μ 3 − i R + a x x + a y y ] , μ = M 2 M 1 + M 2 μ i = m i M 1 + M 2 , δ 1 i 2 = ( x i + 1 ) 2 + y i 2 , ρ 2 i 2 = x i 2 + y i 2 , R 2 = ( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 , n 2 = 1 + 3 A 1 2 , A 1 = R e 2 − R p 2 5 R ¯ 2 .

R_{e} = Equatorial radius of larger mass,

R_{p} = Polar radius of the larger mass,

R ¯ = the mutual distance between the primaries.

The Equation (5.1) may be detailed as

x ¨ 1 − 2 n y ˙ 1 = 1 μ 1 [ n 2 ( 1 − μ + x i ) − ( 1 − μ ) ( 1 + x i ) δ 1 i 3 − μ x i ρ 2 i 3 − 3 ( 1 − μ ) A 1 ( 1 + x 1 i ) 2 r 11 5 + a x 1 − μ 2 R 3 ( x 1 − x 2 ) ]

y ¨ 1 + 2 n x ˙ 1 = 1 μ 1 [ n 2 y 1 − ( 1 − μ ) y 1 δ 1 i 3 − μ y i ρ 2 i 3 − 3 ( 1 − μ ) A 1 y 1 2 δ 11 5 + a y 1 − μ 2 R 3 ( y 1 − y 2 ) ]

x ¨ 2 − 2 n y ˙ 2 = 1 μ 2 [ n 2 ( 1 − μ + x 2 ) − ( 1 − μ ) ( 1 + x 2 ) δ 12 3 − μ x 2 ρ 22 3 − 3 ( 1 − μ ) A 1 ( 1 + x 2 ) δ 12 5 + a x 1 − μ 2 R 3 ( x 1 − x 2 ) ]

y ¨ 2 + 2 n x ˙ 2 = 1 μ 2 [ n 2 y 2 − ( 1 − μ ) y 2 δ 12 3 − μ y 2 ρ 22 3 − 3 ( 1 − μ ) A 1 y 2 2 δ 12 5 + a y 2 − μ 2 R 3 ( y 2 − y 1 ) ]

Then the equilibrium point (x_{10}, y_{10}), (x_{20}, y_{20}) will be given by

a x 1 = − n 2 ( 1 − μ + x 1 ) + ( 1 − μ ) ( 1 + x 1 ) δ 11 3 + μ x 1 ρ 21 3 + 3 ( 1 − μ ) A 1 ( 1 + x 1 ) 2 δ 11 5 + μ 2 ( x 1 − x 2 ) 2 δ 11 3 + μ 2 ( x 1 − x 2 ) R 3

a y 1 = − n 2 y 1 + ( 1 − μ ) y 1 δ 11 3 + μ y 1 ρ 21 3 + 3 ( 1 − μ ) A 1 ( 1 + x 1 ) δ 11 5 + μ 2 ( y 2 − y 1 ) R 3

a x 2 = − n 2 ( 1 − μ + x 2 ) + ( 1 − μ ) ( 1 + x 2 ) δ 12 3 + μ x 2 ρ 22 3 + 3 ( 1 − μ ) ( 1 + x 2 ) δ 12 5 + μ 1 ( x 2 − x 1 ) R 3

a y 2 = − n 2 y 2 + ( 1 − μ ) y 2 δ 12 3 + μ y 2 ρ 22 3 + 3 ( 1 − μ ) y 2 δ 12 5 + μ 1 ( y 2 − y 1 ) R 3

Let us take the variation in the coordinates as in the referred paper

ξ i = x i − x i 0 , η i = y i − y i 0 , p ξ i = p x i − p x i 0 , p η i = p y i − p y i 0 .

And the corresponding Hamiltonian may be written as

H = 1 2 ∑ i = 1 2 [ ( p ξ i 2 + p η i 2 ) + η ( p ξ i η i − p η i ξ i ) + n 2 ξ i ( μ − 1 − x 0 ) − n 2 η i y 0 − 1 − μ δ 1 i − μ ρ 21 − ( 1 − μ ) A 1 2 δ 1 i 3 − 1 2 μ 3 − i R − a x i ξ i − a y i η i ] .

To avoid repetition we shall after some algebraic manipulation with the addition of the extra term − ( 1 − μ ) A 1 2 δ 1 i 3 − 1 2 μ 3 − i R in the Hamiltonian, we may write in dimensional units the minimum distance from the second primary as r 2 i ( min ) ≈ l ( 25 μ ) 1 3 and the required dimensional control acceleration will be a ( r 2 i ( min ) ) = G M 2 μ l 2 [ ( 25 μ ) 1 3 | 1 − 25 μ | | 4 − 25 μ | 2 3 ] 50 . So we find that the minimum

control acceleration is less than the one with the absence of obliquity in the shape of the primary body.

In the present work we have investigated the minimum acceleration that will put two spacecrafts at definite equilibrium position. It is concluded that the presence of a primary with the shape of an oblate spheroid decreases the thrust and it is an advantage over the case when the shape is to be spherical or the bodies are taken to be point masses. Here the two cases of the restricted problem of 2 + 2 bodies have been considered and utilizing this result established by (Whipple, 1984) for negligible masses µ_{1} & µ_{2}, this problem has been split into two separate claimed restricted problems of three bodies and the problem reduces to that studied in the referred paper [

Ranjana, K. and Kumar, V. (2017) On the Artificial Equilibrium Points in the Circular Restricted Problem of 2 + 2 Bodies. International Journal of Astronomy and Astrophysics, 7, 239-247. https://doi.org/10.4236/ijaa.2017.74020