Stability of Collinear Equilibrium Points in Robe s Generalised Restricted Three Body Problem

Transcription

1 Stability of Collinear Equilibrium Points in Robe s Generalised Restrited Three Body Problem K.T. Singh a, B.S. Kushvah b and B. Ishwar (a) Leturer in Mathematis M.B. College, Imphal (Manipur), (b) J.R.F D.S.T. Projet*, () Prinipal Investigator D.S.T. Projet* *University Department of Mathematis, B.R.A. Bihar University Muzaffarpur-,India ishwar_bhola@hotmail.om Abstrat: We have examined the stability of ollinear equilibrium points in Robe s generalized restrited three body problem. The problem is generalised in the sense that more massive primary has been taken as an oblate spheroid. We have found the position of equilibrium points. We have obtained variational equations of the problem. With the help of harateristi roots, we onlude that ollinear points are unstable. Robe s result may be verified from the generalised result. Keywords: Stability/Collinear Equilibrium Points/Robe s Generalised RTBP.. Introdution "Robe's Restrited Three Body Problem", a model onsidered by Robe (77), is a new kind of Restrited Three Body Problem in whih one of the primaries is rigid spherial shell m filled with a homogeneous inompressible fluid of density ρ. The seond primary is a mass point m outside the shell and the third body m 3 is a small solid sphere of density ρ 3 inside the shell with the assumption that the mass and radius of m 3 are infinitesimally small. He has shown the existene of an equilibrium point with m 3 at the entre of the shell while m desribes a Keplerian orbit around it. Further, he has disussed the linear stability of the equilibrium point for the whole range of parameters ourring in the problem. Later on, several authors have studied the Robe's problem. Shrivastava and Garain () have studied the effet of small perturbations in the Coriolis and entrifugal fores on the loation of equilibrium points in the Robe's problem. Plastino and Plastino (5) have onsidered the Robe's problem by taking the shape of the fluid body as Rohes' ellipsoid. They have studied the linear stability of the equilibrium solution too. Giordano, Plastino and Plastino (7) have disussed the effet of drag fore on the existene and stability of the equilibrium points in Robe's problem. In all the above problems, the existene of only one equilibrium point is disussed. Hallan and Rana () established that in the Robe's ellipti restrited three body problem, there is only one equilibrium point while in the Robe's irular restrited three body problem, there are two, three and infinite

2 number of equilibrium points depending upon the ertain values of parameters ourring in the problem. In this paper we have studied the stability of the libration point in Robe's restrited three body problem. The problem is generalised in the sense that bigger primary is taken as an oblate spheroid..stability of Collinear Equilibrium Points The equations of motion in he Robe s generalised restrited three body problem are given as & x& ny&, & y nx&, & z ---- () x y z where, n ( x y ) kr ---- () r ( ) x y, r ( x ) y r z z ---- (3) The angular veloity is given by 3 ρ n A, k π ρ, 3 ρ3 m m m Coordinates of m and m are (-,, ) and (-,, ). The libration points exist when x y z We have found the ollinear libration point at 3 x A k, y, z ---- () At equilibrium points xy yx zx xz ---- (5) and zz k 3 r ---- (6) xx n k ( x ) (7) yy n k x ---- () ( ) 3 Here supersript denotes the orresponding value at equilibrium point

3 Equations (6) and (7) may be written as 3 where B xx A, yy B A A, B n k ( x ) 3 Let the third body of infinitesimal mass be displaed to ( ξ, y η, z ζ ) ξ η, ζ y x where, are very small quantities and ( x ) z is libration point. Using equation (5), the variational equations of motion in the linearized form beomes & ξ n & η xxξ, & η n & ξ yyη, && ζ zz ζ ---- () Again, using equations (6), (7) and () we have ξ& n & η ( B A ξ & ), η& n & ξ ( B A )η &, & ζ ( k A)ζ ---- () Now, we onsider ξ A' e, η B' e, ζ C' e where A ', B', C' are very small quantities and λ is a small parameter. Using the above values in (), we have from 3 rd equation of () λ C' e ( k A) C' e, λ ±i ( k A) whih shows that the motion parallel to z axis is stable. Now, from first two equations of (), we have, s λ B A A' nb' λ { ( )} na' λ λ ( B A) B' { } This will have a non trivial solution for A and B if λ nλ ( B A) nλ λ ( B A) i.e. ( B A n ) λ ( B A)( B A) λ ---- () It is the harateristi equation of problem

4 ( A n ) B ± D λ where D is the disriminate for ritial mass, D or, ( B A n ) ( B A)( B A) on putting the value of B in above equation we have, ( B A n ) ( B AB A ) { } { n k A n } k ie. ( n k A n ) ( n k) A( n k) A ie. ( n k A) ( ) ( ) A ie. A n A 3n k n ( n k) or A ± ---- () If the positive sign of above equation is onsidered then, A n n k A k ---- (3) 3 if negative sign is onsidered A k Now, we know that A ( x ) 3 putting the value of x from equation (), we have, ---- () 3 ( x ) A 3 ie. { ( x )} A ie. ie. 3 A k A 6 A k A 3

5 ie. 5 6 A k A [ A( A k ) ± ---- (5) A k A k ( ) ( ) taking the positive sign in equation (5) ( A k) [ A( A k) ( A k) Using the positive value of A is A [ A ( A k) ( A k) ( A k) Using the value of A (3), we have A k < ---- (6) 3 Similarly using the value of A - in plae of A, we get [ A ( A k) ( A k) ( A k) Using the value of A - from () < ---- (7) k Again taking negative sign in (6) ( A k) [ A( A k) ( A k) Using the value of A in plae of A in the above equation [ A ( A k) ( A k) ( A k) A k ( A k) () Again using the value of A - in the equation in plae of A

6 6 A ( ) A k ( A k) ( A k) k A k ---- () ( ) whih is a very small quantity and liable to be negleted. Thus we get the admissible value of ritial mass as dedued in equation (6s) A k 3 Putting A & k we get whih agrees with the result of Robe (77) the (-) sign appears beause the value of lies on the negative side of origin in the x axis as investigated in the orientation of the onfiguration. Thus A k where and at A & k 3 we have, - y -x x A y' If < or > or < we have a range of stability And if < or we have a range of instability. or

7 Referenes 7 [.. Hallan, P.P. and Neelam Rana (): The existene and stability of equilibrium points in the Robe s restrited three body problem. Celest. Meh. & Astr. Dyn., [. Kumar, V. and Choudhry, R.K. (7): On the stability of the triangular libration points for the photogravitational irular restrited problem of three bodies when both the attrating bodies are radiating as well. Indian J. Pure Appl. Math. (), 3-3. [3. Liapunov, A.M. (56): A geneal problem of stability of motion Aad. S. USSR. [. Markellos, V.V., Perdios, E. and Labropoulou, P. (): Linear stability of the triangular equilirbium points in the photogravitational ellipti restrited problem I. Astrophy. and Spae Si.,, 7-3. [5. M Cuskey, S.W. (63): Introdution to Celestial Mehanis, Addison-Wesley Publishing Company, In., New York. [6. Mishra, P. and Ishwar, B. (5): Seond order normalization in the generalized restrited problem of three bodies, small primary being an oblate spheroid. Bull. Astr. So. (India) 3, [7. Robe H.A.G. (7): A new kind of three body problem. Celestial Mehanis, 6, ss [. Sahoo, S.K. and Ishwar, B. (): Stability of ollinear equilibrium points in the generalised photogravitational ellipti restrited three body problem. Bull. Astr. So. (India), [. Sharma, R.K. (7): The linear stability of libration points of the photogravitational restrited three body problem when the smaller primary is an oblate spheroid. Astrophy. and Spae Si. 35, 7-. [. Shrivastaba, A. K. and Garain, D. (): Effet of perturbation on the loation of libration point in the Robe restrited problem of three bodies () Celest. Meh. & Dyn. Astr. 5, [. Singh, J. and Ishwar, B. (): sstability of triangular points in the generalized photogravitational restrited three body problem. Bull. Astr. So. (India) 7, 5-.