Technology, Dhauj, Faridabad Technology, Dhauj, Faridabad

Transcription

1 STABILITY OF THE NON-COLLINEAR LIBRATION POINT L 4 IN THE RESTRICTED THREE BODY PROBLEM WHEN BOTH THE PRIMARIES ARE UNIFORM CIRCULAR CYLINDERS WITH EQUAL MASS M. Javed Idrisi, M. Imran, and Z. A. Taqvi 3,3 Department f Applied Sciences & Humanities, Al-Falah Schl f Engineering & Technlgy, Dhauj, Faridabad mjavedidrisi@gmail.cm Department f Applied Sciences & Humanities, Brwn Hills Cllege f Engineering & Technlgy, Dhauj, Faridabad mhammad.imran85@gmail.cm Received March 03; accepted 9 May 03 ABSTRACT In this paper we have cnsidered the restricted three bdy prblem in which bth the primaries are unifrm circular cylinders having equal mass. We have determined the equatins f mtin f the infinitesimal mass and then we have investigated the cllinear and nn-cllinear statinary slutins and the linear stability f the nn-cllinear libratin pint L 4. Keywrds: Restricted three bdy prblem, Cllinear and Nn-cllinear Statinary slutins, Stability INTRODUCTION The restricted prblem f three bdies is said t describe the mtin f the infinitesimal mass which mves in the plane frmed by the tw primaries. The primaries are revlve arund their center f mass in the circular rbits under the influence f their mutual gravitatinal attractin and the infinitesimal mass is attracted by the primaries but nt influencing their mtin. In the classical case f the restricted three bdy prblem, five libratin pints eist ut f which tw pints are nn-cllinear and three are cllinear.the cllinear libratin pints L, L and L 3 are unstable fr 0 µ ½,while the tw equilateral libratin pints L 4, L 5 are stable fr the critical value f mass parameter µ < µ c = , Szehebely, (967a). This paper is the etensin f A mathematical mdel n the R3BP when the primaries are unifrm circular cylinders by M. Javed Idrisi et.al (03). In this paper we have cnsidered bth the primaries as unifrm circular cylinders with equal mass. Our aim is t investigate the equatins f mtin, statinary slutins and the linear stability f the nn-cllinear libratin pint L 4. Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

2 M. Javed Idrisi et al. EQUATIONS OF MOTION Figure : The Cnfiguratin f the R3BP when bth the primaries with mass m and m are unifrm circular cylinders Let m and m (m = m ) be unifrm circular cylinders f same length and radius l and κ respectively. They are mving in the circular rbits arund their center f mass O. An infinitesimal mass m 3 having c-rdinates (, y) mving in the plane f mtin f m and m. The distance f the infinitesimal mass m 3 frm m, m and O is r, r and r respectively. We wish t find the equatins f mtin f m 3 using the terminlgy f Szehebely (967), in syndic system and dimensinless variables. The ptential f the cylinder f mass M f length l and at a distance r frm P(,y) is given by V = M (lg l lg r) () (Ramsey, 959) where M = mass f the cylinder, l = length f the cylinder and r = the distance between the center f mass f the cylinder and the pint P (at which we have t find the ptential f the cylinder). Let the ptential f the cylinders m and m be V and V respectively. Using the terminlgy f Szehebely the ptential f the primaries f m and m at the pint P(, y) in dimensinless variables is therefre given by V = lg l lg r () V = lg l lg r (3) Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

3 Stability f the nn-cllinear libratin pint L 4 3 where r = + y, r = + + y, m = m =.. The equatins f mtin f m 3 in syndic system and dimensinless variables are: ny = n (4), r r y+ n = n + + y. (5) r r where n is the mean-mtin f the primaries. Nw we define a functin Ω such that n Ω= ( + y ) ( V+ V) (6) where V and V are defined in the Equatin () and the Equatin (3). Hence, the Equatins (4) and (5) becme =Ω ny y + n =Ω y (7) (8) The integral analgus t Jacbi integral is + y = v = Ω C (9) ( ) where v is the velcity f m 3 and C is the cnstant f integratin.. Mean-mtin f the primaries The gravitatinal ptential f tw bdies f finite dimensin is given by Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

4 4 M. Javed Idrisi et al. fmm' fm' fm U = + 3 ( A+ B+ C 3 I ) + 3 ( A ' + B ' + C ' 3 I ') (0) r r r where f = cnstant f gravitatin, M and M = masses f the finite bdies, r = distance between the centre f mass f M and M, A, B and C = mment f inertia f M abut the principal aes, A, B and C = mment f inertia f M abut the principal aes, (Bruwer and Clemence, 96) I = mment f inertia f M abut the line jining the center f mass f M and M, I = mment f inertia f M abut the line jining the centre f mass f M and M. Since the center f mass f M and M lies n the -ais therefre I = A and I = A. Thus the gravitatinal ptential between tw cylinders f masses m and m is given by fmm fm fm U = + ( B+ C A) + ( B' + C' A') () 3 3 r' r' r' where r = distance between m and m. On calculating the values f A, B, C and A, B and C, the Equatin () becmes 8l 6κ U= fmm + r 3 ' r ' () Therefre the mean mtin f the primaries is given by 4l 3κ n = +. (3) 6 If κ =, n = which is the classical case f the restricted three bdy prblem. 4l 3 0 Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

5 Stability f the nn-cllinear libratin pint L STATIONARY SOLUTIONS The Libratin pints are the singularities f the manifld F yy + y Ω+ C= (,,, ) 0 and these pints are given by the Equatins F = F = F = F = 0 y y ie.. = 0, y = 0, Ω = 0, Ω = 0 We have y Ω = = 0, n r r (4) Ωy y n + + = r r 0. (5) Frm the Equatin (5) we have tw cases, either y = 0r y Cllinear Slutins Case I: When y = 0 The Equatin (4) becmes n+ + = 0, + (6) The Equatin (6) is a third degree equatin in. On simplifying the Equatin (6) we get n n = 0 (7) On slving the Equatin (7) we have n 8 = 0 and = ± (8) n Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

6 6 M. Javed Idrisi et al. It is clear frm the Equatin (8), there eist three cllinear Libratin pints L = 0, n 8 L,3 = ±. The secnd and third Libratin pints L and L 3 eist if and nly if n 3 i.e. n 3 l + κ Triangular Slutins Case II: When y 0 The Equatin (5) becmes n + 0 r + r = (9) The Equatin (4) can be written as n = r r r r (0) Using the Equatin (9) we have = 0 () r r The slutin f the Equatin () is r = r. If r = r, the infinitesimal mass m 3 frm an issceles triangle with base m and m. Figure : Lcatin f the nn-cllinear Libratin pint L 4 Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

7 Stability f the nn-cllinear libratin pint L 4 7 Let r = r = p. Therefre, the c-rdinate f L 4,5 is given by 0, ± p, p> 4 () If p =, the infinitesimal mass m 3 frm an equilateral triangle with base m and m and the crdinate f L 4 is 3 0, ±. 4 STABILITY OF THE LIBRATION POINTS The Equatins f the mtin f the infinitesimal mass are ny =Ω y + n =Ω y (3) Let the c-rdinates f libratin pint be ( y.we, ) give small displacement (α, β) t ( y., ) The equatins f mtin becme α n β =Ω ( + α, y+ β) = αω + βω β + n α =Ω ( + α, y+ β) = αω y + βω y y yy (4) where dentes that the partial derivatives are t be calculated at the Libratin pint under cnsideratin. The characteristic Equatin f (4) is given by 4 n yy λ yy y λ + (4 Ω Ω ) + Ω Ω ( Ω ) = 0 (5) which is a furth degree equatin in λ. The Libratin-pint ( y, ) is stable if all the fur rts f the Equatin (5) are pure imaginary. On differentiating the Equatin (4) with respect t and y, we get Ω = , n 4 4 r r r r (6) Ω y = y, r r (7) Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

8 8 M. Javed Idrisi et al. On differentiating the Equatin (5) with respect t y, we get Ω yy = n + + y r r r r (8) 5. Stability f the Nn-cllinear Libratin pint L 4 The c-rdinates f the nn cllinear libratin pint L 4,5 is 0, ± p, p>. 4 Therefre at the Libratin pint L 4,, p p Ω = n + 4 Ω y = 0,. p p Ω yy = n + 4 Thus, the characteristic Equatin (5) becmes λ + n λ + n + = 0 (9) p p p On slving the Equatin (9) we have λ p n p +, λ p =± =± n p (30) p p, 3,4 Fr p > and n, all the rts f the Equatin (9) (λ i, i =,,,4) are pure imaginary. Thus, the Libratin pint L 4 is stable. 6 CONCLUSION In the classical case f the restricted three bdy prblem the mean mtin f the primaries is unity but in ur case the mean mtin n depends upn the lengths and radii f the cylindrical primaries [Equatin (9)]. In the classical case there eist three cllinear and tw nncllinear pints. All the cllinear Libratin pints are unstable but the nn-cllinear Libratin pint is stable fr µ < In ur case there als eist three cllinear and tw nn cllinear pints and the nn-cllinear pint L 4 is stable fr p > and n [Equatin-(30)]. Int. J. f Appl. Math and Mech. 9 (5): -9, 03.

9 Stability f the nn-cllinear libratin pint L 4 9 REFERENCES Bruwer D and Clemence G M (96): Methds f Celestial Mechanics, Academic Press, New Yrk and Lndn, p Idrisi M Javed, Imran M and Taqvi Z A (03): Internatinal Jurnal f Applied Mathematics and Mechanics, vl. 9 (5), p McCuskey S W (963): Intrductin t Celestial Mechanics, Addisn-Wesley Publishing Cmpany, Inc., United States f America. Ramsey A S (959): An Intrductin t the Thery f Newtnian Attractin, Cambridge, p Szebehly Victr (967a): Thery f Orbits, The Restricted Prblem f Three Bdies, Academic Press, New Yrk and Lndn. Int. J. f Appl. Math and Mech. 9 (5): -9, 03.