Triangular Libration Points in Elliptic Photogravitational Restricted Three Body Problem with Oblateness

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 9, Issue 1, January 2019)

11 Triangular Libration Points in Elliptic Photogravitational

Restricted Three Body Problem with Oblateness

Dr. Ganesh Mahto

Associate Professor of Mathematics, L. N. M. U., Darbhanga, (Bihar) India

Abstract:-- This paper investigates the positions of triangular libration points (L4,5) in the elliptic restricted three-body problem (ER3BP), when one is oblate primary and other emit light energy. The positions of the triangularpoints are seen to shifted away from the line joining the primaries than in the classical case due to the effect of the introduction of the eccentricity, semi-major axis, radiation and oblateness factors.

Keywords:-- Triangular Libration Points, Photogravitational, Oblatenes, ER3BP.

I. INTRODUCTION

The elliptic restricted three-body problem (ER3BP) describes the three-dimensional motion of a small particle, called the third body (infinitesimal mass) under the gravitational attraction force of two finite bodies, called the primaries, which revolve on elliptic orbits in a plane around their common centre of mass. The circular restricted three-body problem (CR3BP)describes with first approximation, the actual situation of an infinitesimal mass in the gravitational field of the primaries moving in circular orbits. The major fault of the CR3BP in celestial mechanics is its instability to treat the along time behavior of practically important dynamical systems. The principal reason is that significant effects might be expected because of the eccentricity of the orbits of the primaries. The orbits of most celestial bodies are elliptic rather than circular, as such, the ER3BP analysis the dynamical system more accurately.

The restricted three-body problem is unable to discuss the motion of the infinitesimal body when at least one of the participating bodies is an intense emitter of radiation. The photogravitational CR3BP formulated by Radzievsky (1950) may be the simplest model for this problem. Among the various possible motion of the particle, the equilibrium positions around of a rotating (together with the stars) system of coordinates have practical applications. The photogravitational restricted three-body problem models adequately, the motion of a particle of a gas-dust cloud which is in the field of two gravitational and radiating stars. The summary action of gravitational and light repulsive forces may be characterized by the mass reduction factor q.

The existence and stability of equilibrium points were studied by Chernikov (1970) in the case when only one body radiates, while Schuerman (1980), Luk’yanov (1984), 1988), Simmons et al. (1985) and Kumitsyn (2000, 2001) in the case when both bodies are luminous.

The rotation of a star produces an equatorial bulge due to centrifugal force; as a result, the stars are often oblate in shape. The bodies in the classical restricted three-body problem are strictly spherical, but some planets (Earth, Jupitor and Saturn) and stars (Archerner, Antares and Altair) are sufficiently oblate to make the departure from sphericity very significant in the study of celestial and stellar systems. Taking one or both primaries as sources of radiation or oblate spheroids or both, the ffects of oblateness and radiation pressure of the primaries on the existence and stability of equilibrium points in the CR3BP were analyzed by Singh and Ishwar (1999), Ishwar and Kushvah (2006).

The ER3BP generalizes the original CR3BP, and improves its applicability, while some outstanding and useful properties of the circular model still hold true or can be adopted to the elliptic case. In particular, possible positions of equilibrium occur when the three bodies from equilateral triangles. An application of this model can be seen in the motion of the Trojan asteroids around the triangular point L4. The asteroids in this case are only influenced by the gravitational forces of the Sun and Jupiter, and the orbit of Jupiter around the Sun is assumed to be a fixed ellipse. The influence of the eccentricity of the orbits of the primary bodies with or without radiation pressure(s) on the existence of the equilibrium points and their stability was touched upon to some extent by Kumar and Choudry (1990), Markellos et al. (1992), and Ammar (2008).

The present study aims to examine the motion of the infinitesimal body in the ER3BP when one primary is sources of radiation and other oblate spheroid, which will contribute a lot to understanding the effects of radiation, eccentricity and oblateness on the celestial and stellar systems.

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International Journal of Emerging Technology and Advanced Engineering

12

II. EQUATIONS OF MOTION

We consider two bodies (primaries) of masses

m

₁and

2

m

with

m

₁



m

₂moving in a plane around their common

center of mass in elliptic orbit and a third body (infinitesimal mass) of mass m is moving in a plane of motion of the primaries. Equations of motion of our problem in rotating and pulsating co-ordinate system are given by (Sahoo and Ishwar 2000)

x

y

x















2

(1)

y

x

y















2

(2)

z











(3)



 











































2 2 3

1 1

1 2 2 2

2

₂

1

2

1

1 r

q

r

A

r

n

y

x

e



(4)

The mean motion of our problem is obtained as The angular velocity



is defined as



 





₂



32

2 3

2 2

1 2 1 2

1 cos

1 e

a

e

m

K











Szebehely (1967)

where

K

2is gravitational constant,

a

is the semimajor axis of the ellipse,

e

is the eccentricity of the ellipse,



is the true anomaly.

Since the value of



is concerned with



, so it varies with the positions of the primaries. Now the average of the angular velocity i.e., the mean motion is



 





















2

0 32 ₂ 32

2 2

1 2 1 2

1 cos

1

2

1 d

e

a

e

m

K

n



















d

e

a

m

K











2

0

2

2 3 2 2

3

2 1 2 1 2

cos

1

2

(3)

International Journal of Emerging Technology and Advanced Engineering

13

Now



















































 





























2

1

2

2 sin

2 cos

1

2 cos

2

1 cos

cos

2

1 cos

1

2

0 2

2

0

2 2

2

0

2 2 2

0

2

e

d

e

d

e

d

e





 

Substituting the value of the above integral in equation (5), we have











₂



32

2 3

2 2 1 2 1

1

2

1 e

a

e

m

K

n







So the mean motion is

K



m

₁



m

₂



12

a

32 in circular case. In elliptic case, the distance of the two primaries is given by







cos

1

2

e

a

r







The mean distance of the two primaries is given by











2

0

2

1 rd







_

_









2

0 2

cos

1

2

1 e

d

e

a



_{ }

_













2

0

4 4 3 3 2 2 2

...

cos

1

2

1 d

e

a











2

0 2

2

...

2

2 sin

2

1 





















 





a

e





















...

8

3

2

1

2

1 e

2

e

2

e

4

a

_





₂



₂



12

1

1 





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International Journal of Emerging Technology and Advanced Engineering

14 





₂



12

2

1

1 e

e

a







0 ,

0 



r



Now





F

n

e

a

r











2

2 2

cos

1

1 





The orbit of mass

m

₁with respect to the centre of mass with semi major axis

a

₁



m

₁

a

and similarly, the orbit of

m

₂with respect to the centre of mass with semi major axis

a

₂



m

₂

a

(Szebhely 1967).

Then



















 







2

3

1

₁

2 2 2

1 2

2 1 2

A

m

K

e

am

n

(6)



















 







2

3

1

₁

1 2 2

1 2

2 2 2

A

m

K

e

am

n

(7)

Considering the distance between the two primaries as unity and adding the expressions (6) and (7), we have



























 











2

3

1

₁

2 1 2 2

1 2

2 2

1 2

A

m

K

e

a

m

n



















 







2

3

1

2 1

2 1 2

2 2

A

K

e

a

n

Since

K

2



1 

2



2 1

2

1

2

3

1 e

a

e

A

n















 



(8)

a

is semi-major axis of the ellipse

e

is the eccentricity of the ellipse

1

A

is the coefficients of oblateness of bigger primary







2



2



2



1 ,

2 

x

y

z

i

r

_i _i (9)

2 1

2 2

1

,

1 ,

m

x











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International Journal of Emerging Technology and Advanced Engineering

15

Here

m

₁,

m

₂are the masses of the bigger and smaller primaries (x1,0,0) and (x2,0,0) are the coordinate of

m

1and

2

m

respectively.

q

₁ is mass reduction factor.

2 2 2

1

5 r

r

A



e



p is oblateness coefficient due to bigger

primary

m

₁, where

r

_e

,

r

_p represents equatorial radii and polar radii respectively.

r

_i



i



1 ,

2 

are the distance of the infinitesimal mass from

m

₁and

m

₂ respectively. Semi-major axis and eccentricity of orbit are denoted by

a

and

e

respectively.

Now multiplying equations (1), (2) and (3) by

2 x



,

2 y



and

2 z



respectively and adding we get



x

F

x

y

F

y



dt

dC

_

_

_

_

_

2

(11)

where

C



2 



x



2



y



2, the quantity C is Jacobi integral. The zero velocity curves are given by

C



2 

.

III. LOCATIONS OF THE LIBRATION POINTS

Thelibration points are the solutions of the equations

0 











_x _y _z .

that is,









 





0

2

1

3

1

5 2

1 3

2 2 3

1

2



















_





_







r

A

x

r

x

q

r

x

n

x











0

2

1

3

1

₅

1 1 3

2 2 3

1

2





























_



r

A

r

q

r

n

y











0

2

1

3

1

5 1

1 3

2 2 3

1

















_



r

A

r

q

r

z



(12)

The solutions of the first two equations of systems (12) with

y



0 ,

x



0

give the positions of the triangular points. From which we obtain

3 1

1 3 1 2

2

3

1 r

A

r

n





, ₃

2 2 2

r

q

n



(13)

with ₂3 2₂

n

q

r



when oblateness of the smaller primary is absent.Thisvalues changes slightly by



₁(say), with the introduction of oblateness, so hence

1 3 2 1

1 _

_



n

r

, ₂₃

3 1 2 2

n

q

r





₁



1

(14)

Considering only terms in A1 and e2 and neglecting their products gives













_



2

1 2

2

3

2

3

1

1 e

A

a

n

(15)

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International Journal of Emerging Technology and Advanced Engineering

16  

 

_

































2

1

2

1

2 3 1 2 2

1 2 3 1 1

e

aq

r

e

a

r



(16)

From (13), (14) and (15) and neglecting higher order terms in



₁

,

A

₁

,

e

2, we get

 

_

_{ }

23

_

1

1 3 1

1

2





a

A

a



(17)

Substituting for



₁in (16), we obtain

 



 



 



1



2 3 2 2 2 2

3 2 1 1 2 3 2 2 1

1

1 A

e

aq

r

a

A

e

a

r













(18)

Using (18) and equations (9), (10), we get

 









 



 







 









 





 



 





 



1





12

2 3 2 2 3

2 1 1 2

3 2 3

2 1 1 2 3 2

1 2 3 2 2 3

2 1 1 2 3 2

1

2

1

2

1

4

1

2

1

2

1 A

e

aq

a

A

e

a

A

e

a

y

A

e

aq

a

A

e

a

x































 





(19)

The triangular libration points denoted by L4,5



x

,



y



are given by (19).

IV. CONCLUSION

We have found the position of triangular libration points in ER3BP in which one primary is radiating and other oblate. We conclude that their positions are significantly affected by the eccentricity of orbits, oblateness and radiation factors of the primaries. All classical results may be verified.

REFERENCES

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[4] Ishwar, B., Kushvah, B. S. : J. Dyn. Syst. Geom. Theories 4, 1 (2006).

[5] Kumar, S., Ishwar, B: AIP Conf. Proc. 1146, 456 (2009).

[6] Kunitsyn, A. L.: J. Appl. Math. Mech. 64, 757 (2000). [7] Kunitsyn, A. L.: J. Appl. Math. Mech. 65, 703 (2001). [8] Kumar, V., Choudry, R. K. : Celest. Mech. 48, 299 (1990). [9] Luck’yanov, L. G. : Astron, Zh. 65(2), 422-432 (1988), Sov. Astron.

32, 6.

[10] Markellos, V. V., Perdios, E., Labropoulou, P. : Astrophys. Space Sci. 194, 207 (1992).

[11] Schuerman, D. W.: Astrophys. J. 238, 337-342 (1980).

[12] Simmons, J. F. L., McDonald, A. J. C. Brown, J. C.: Celest. Mech. 5, 145 (1985).

[13] Singh, J. Ishwar, B. : Bull. Astron. Soc. India 27, 415 (1999). [14] Szebehely, V. G. : Theory of Orbits. Academic Press, New York

(1967).