Microsoft Word - R.R. Thapa _121-125_.doc R. R. Thapa and M. K. Raut. / BIBECHANA 9 (2013) 121-125: BMHSS, p.121 (Online Publication: Nov., 2012) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA The sensitivity of L4 and L5 in the framework of restricted problem of three bodies when the primaries are different R. R. Thapa, M. K. Raut Department of Mathematics, P.G. Campus, Biratnagar, Nepal Article history: Received 2 October, 2012; Accepted 16 November, 2012 Abstract In the paper, the location of liberation points have been observed and triangular liberation points L4 and L5 are determined. Keywords: Libration points; Jacobi integral; triaxial rigid body 1. Introduction Two primaries the point masses, the bigger one is spherical and the second primary; (smaller primary) is a triaxial rigid body are moving around their centre of mass in circular orbits under the influence of their mutual gravitational attraction and a third body, influenced by the primaries but not influencing their motion moves in the plane defined by the two revolving primaries. If one body is triaxial and other is spherical then the triangular libration points in restricted problem of three bodies are infinitesimal or linearly stable. Leontovic [1] established a non-linear stability of the triangular libration point. Deprit and Palmore [2] established the family of short orbits originating at the equilateral triangular libration point. Deprit [2,3] studied geometrically the long periodic and short periodic obits around L4 with the help of D'Alembert's series. Henrard [4] discovered that the long periodic orbits at L4 doesn't evolve in a continuous way. And discontinuity appears not only at mass ratios but also at singular bifurcation points. Tuckness [6] has used all the stability criteria numerically and investigated the sensitivities of third body around L4 when it is given positional deviations away from L4 with a suitable condition. The above mentioned mathematicians studied the different aspects of stability of libration points with different approaches in circular restricted problem of three bodies. Here we have taken the bigger primary is a spherical and smaller as a triaxial rigid body under the same condition of Mc kanzie and Szebehely and Tukness. 2. Equation of Motion The equations of motion of the third body are: R. R. Thapa and M. K. Raut. / BIBECHANA 9 (2013) 121-125: BMHSS, p.122(Online Publication: Nov., 2012) x y2x ∂ Ω∂ =η− ••• y x2y ∂ Ω∂ =η+ ••• where ( )22 2 1r x y= − µ + (2) ( )22 2 2r x 1 y= − µ + + (3) and ( ) ( ) 2 1 22 2 2 1 2 1 23 5 1 2 2 2 21 3 (1 )r r y 2 r r 2r 2r µ σ − ση − µ µ µ  Ω = + µ + µ + + + − σ − σ  (4) 21 21 2 , 2 1 mm mm m ≥≤ + =µ Where r1,r2 are distances between first and second primary from third body respectively, and m1,m2 are masses of the first and second primaries respectively. The mean angular motion, ( )212 2 3 1 σση −+= (5) where , 5 , 5 , 5 2 2 32 2 22 2 1322311 R c A R b A R a AandAAandAA ===−=−= σσ a, b, c being semi-axes of triaxial rigid body and R is distance between the primaries. 3. Location of Libration Points From equation (1) ( )yx dt d 2y y x x 2yx dt d 22 ,.. Ω=      ∂ Ω∂ + ∂ Ω∂ =      + •••• Integrating w. r. t. t we get c2yx 22 −Ω=+ •• (6) which is Jacobi's integral. Libration points are solutions of the equations ,0,0. = ∂ Ω∂ = ∂ Ω∂ yx we get, 2 2 1 1 , r y y r r y y r = ∂ ∂ = ∂ ∂ or, ( )( ) ( ) ( ) ( ) ( ) ( ) 2 7 2 21 5 2 21 3 2 3 1 2 1 2 15 1 2 2311 . yx r x rr x r x x x +− − ++− − − +− − −− −= ∂ Ω∂ µ σσµ µ σσµµµµµ η (1) R. R. Thapa and M. K. Raut. / BIBECHANA 9 (2013) 121-125 : BMHSS, p.123 (Online Publication: Nov., 2012) ( ) ( ) ( )       − + − −− − −= ∂ Ω∂ 2 7 2 21 5 2 21 3 2 3 1 2 2 15 2 3431 . y rrrr y y σσµσσµµµ η 4. Triangular Libration Points The triangular libration points are solutions of the equations: ( )( ) ( ) ( ) ( ) ( ) ( ) 01 2 15 1 2 2311 2 7 2 21 5 2 21 3 2 3 1 2 =+− − ++− − − +− − −− − yx r x rr x r x x µ σσµ µ σσµµµµµ η (7) ( ) ( ) ( ) 0 2 15 2 3431 2 7 2 21 5 2 21 3 2 3 1 2 =      − + − −− − − y rrrr y σσµσσµµµ η (8) Multiplying equation (7) by ( )1x +µ− and subtracting the result from equation (8); we set. ( ) ( ) ( ) ( ) 01 31 1 5 2 21 3 1 2 =+− − + − +−− µ σσµµ µη x rr (9) Multiplying equation 8 by ( )µ−x and subtracting the result from equation (7) we get, ( )( ) ( ) ( ) 0 2 15 2 2331 2 7 2 21 5 2 21 5 2 21 3 2 2 =      − + − − −− +− y rrr x r σσσσµσσ η (10) If we put 021 ==σσ then equations ( 9) and (10) become 3 2 2 1 r −η =0 and 0 1 3 1 2 =+− r η Adding them, we get, 21 rr = (11) Using equation 5, we get, 121 === ηrr If 0,0 21 ≠≠ σσ then we get αβµ −+−= 2 1 x (12) ( )    ++±= βα 3 2 1 2 3 y (13) R. R. Thapa and M. K. Raut. / BIBECHANA 9 (2013) 121-125 : BMHSS, p.124 (Online Publication: Nov., 2012) where 1,0,1,1 21 <<<+=+= βαβα rr Putting the values of yxrr ,,,, 21η in equations (9) and (11) We get, ( ) ( ) ( ) ( ) ( )( ) ++−−+      −−−++−−+ −−− 5 21 5 21 3 21 12 2 3 1 2 1 312 2 3 1 βσσβαβσσβσσ ( )( ) 7 21 1 2 15 −+− βσσ 0 4 3 =      ++ βα and ( ) ( )( ) ( ) 01 2 1 )(3111 2 3 31 5 21 3 21 =+      −+−++−+−      −+− −− βαβσσµαµµσσ We get 21 8 11 8 11 σσβ +−= and ( ) ( ) 21 12 21 12 32 σ µ µ σ µ µ α       − − +      − +− = respectively From equation (12) ( ) ( ) 21 18 37 18 3 2 1 σ µ µ σ µ µ µ       − − +      − + −−=x and from equation (13) ( ) ( )       − − + − − −±= 21 112 1915 112 2319 1 2 3 σ µ µ σ µ µ y Thus, ( )             ++−+−≡ βααβµ 3 2 1 2 3 , 2 1 4L and ( )             ++−−+−≡ βααβµ 3 2 1 2 3 , 2 1 5L can be written as ( ) ( ) ( ) ( ) ( ) ( )             − − + − − − − − + − + −−≡ 21214 112 1915 112 2319 1 2 3 , 18 37 18 3 2 1 σ µ µ σ µ µ σ µ µ σ µ µ µL ( ) ( ) ( ) ( ) ( ) ( )             − − + − − −− − − + − + −−≡ 21215 112 1915 112 2319 1 2 3 , 18 37 18 3 2 1 σ µ µ σ µ µ σ µ µ σ µ µ µL 5. Conclusion The triaxial rigid body can be affected by analytical effect on the values of the critical mass. The libration points L4 & L5 are depending on µ , 21 ,σσ . R. R. Thapa and M. K. Raut. / BIBECHANA 9 (2013) 121-125: BMHSS, p.125 (Online Publication: Nov., 2012) References [1] A. M. Leontovic, On the stability of the Lagrange periodic solutions for the reduced problem of three bodies, Soviet Math, 3(1962) 425 - 428. [2] A. Deprit and Palmore, Analytical continuation and first order stability of short period orbits at L4, Astron, J.,71(1966 a) 94. [3] A. Deprit, Limiting orbits around the equilateral centres of libration, Astron, J., 71(1966b) 77 - 87. [4] J. Henrard, Concerning the Genealogy of long period families at L4 , Astron, Astrophysics, 5(1970) 45-52.