Ameicn Jounl of Applied Sciences 9 (9: 78-8, ISSN 56-99 Science Publiction Discovey of n Equilibium Cicle in the Cicul Resticted Thee Body Poblem, Fwzy A. Abd El-Slm Deptment of Mth, Fculty of Science, Tibh Univesity, Al-Mdeenh, Sudi Abi Deptment of Astonomy, Fculty of Science, Cio Univesity, Cio, 6, Egypt Abstct: Poblem sttement: A til to find equilibium points in out of plne of the esticted thee body poblem. Appoch: Solution of the equtions of motion t equilibium points. Lineizing the equtions of motion nd computing the eigen vlues to investigte the stbility. Results: New tingul equilibium points in plne pependicul to the plne of motion nd pssing though Lgnge tingul equilibium points e obtined. A cicle of equilibium points, nmely Fwzy equilibium cicle is discoveed. Infinite numbe of equilibium points locted on the cicumfeence of this cicle e computed. The obtined solutions e checked vi obtining some specil cses. The stbility of Fwzy equilibium cicle is studied. The oscilltoy stble solutions s η-dependent nd ξ- dependent stbilities e deived. Conclusion: The equilibium points in out of plne of motion of the esticted thee body poblem is investigted. We obtined the following vey new esults:- Fist we obtined Fwzy ξξ-tingul equilibium points in the plne η =. Second we obtined the so clled Fwzy equilibium cicle. We checked ou solutions vi obtining some specil cses. We studied the stbility of Fwzy equilibium cicle. We deived the oscilltoy stble solutions s η-dependent nd ξ- dependent stbilities. The stbility of Lgnge s well s Fwzy tingul equilibium points followed diectly becuse they e subsets of the Fwzy equilibium cicle. Key wods: Fwzy equilibium cicle, stbility, the thee body poblem INTRODUCTION In ny ssumed isolted two-body mssive obiting system (such s the Sun nd the Eth thee e five equilibium points, L i, i =,,,, 5 these points usully clled Lgngin o libtion points. At these points the gvittionl pulls e in blnce. Any infinitesiml body t ny point of the Lgngin points would be held thee without getting pulled close to eithe of mssive bodies. The points L, L, L e coline with the line joining the two mssive bodies, while the tingul points L, L 5 e found 6 hed of nd behind the less mssive body, long its obit. These two tingul points L,5 e foming equiltel tingles with the two mssive bodies. The Resticted Thee Body Poblem (RTBP in bief is now defined s system consisting of two mssive bodies, clled the pimies, evolving in cicul obits ound thei cente of mss nd thid body of infinitesimlly smll mss which moves in the pimies' obitl plne. The thee colline equilibi t L, L, L e unstble, while the two tingul solutions L, L 5 e stble, in the line nlysis, fo only cetin vlues of the mss tio in the intevl (, µ whee µ =.85 is the petubtions (cf. Szebehely, 967; Subb nd Shm, 99; 997; 988; 986; Nyn nd Rmesh,. The globl stbility of these points hve been studied by sevel uthos: (Leontovich, 96; Depit nd Depit-Btholome, 967; Mkeev, 969; Szebehely 979; Nyn nd Rmesh, 8; Singh, ; Kum nd Ishw, ; Douskos, ; Shnkn et l.,. Thei finl conclusions e tht in the pln cse the tingul points L,5 e lwys stble within some domin of mss tio which is modified when including such kind of diffeent petubtions. In this study the utho is seching fo new equiliium points in the peipindicul plnes to the plne of motion of the pimies. We follow the sme vey well known wy in RTBP but with the equied modifictions. We fist compute thei positions of such equilibium points nd then study thei stbility. Equtions of motion: The equtions of motion of n infinitesiml mss in the eltivistic RTBP in synodic fme of efeence (ξ,η,ξ, in which the pimies coodintes on the x-xis (-µ,,, (-µ,, e kept fixed nd the oigin t the cente of mss, e given Eq. Routhin which modifies when including diffeent by Bumbeg (97: Coesponding Autho: Fwzy A. Abd El-Slm, Deptment of Mth, Fculty of Science, Tibh Univesity Al-Mdeenh, KSA 78
Am. J. Applied Sci., 9 (9: 78-8, U d U ξ n η =. ξ dt ξ U d U η+ n ξ =. η dt η.. U ζ = ζ ( whee n= is the men motion of the otting system nd U is the potentil-like (pseudopotentil function of the RTBP is given by Eq. : µ µ U = + + ( And: ( ξ + µ ( ξ + µ = + η + ζ, + η + ζ =, = ξ + η + ζ Equilibium points: Fom the equtions of motion (, it is ppent n equilibium solution exists eltive to the otting fme when the ptil deivtives of the pseudopotentil function e ll zeo, i.e., these points coespond to the positions in the otting fme t which the gvittionl foces nd the centifugl foce ssocited with the ottion of the synodic fme ll cncel, with the esult tht pticle locted t one of these points ppes sttiony in the synodic fme. The libtion points e obtined fom equtions of motion ( fte setting ξ ξ = = η = η = ζ = ζ =. These points epesent pticul solutions of equtions of motion Eq. : ewite the potentil-like function U in diffeent fom. By definition, this new set hs η = nd Eq. : = ξ + µ + ζ ( = ξ + µ + ζ Thus we cn wite Eq. 5: ( µ ξ + µ + ζ + µ ξ + µ + ζ = ξ + ζ µ µ U = ( µ + + µ + + µ ( µ The system ( cn be witten s Eq. 6: (5 U ( µ µ ξ ξ = = (6 U ( µ µ ζ ζ This system is veified when: µ =, = µ fom which one cn esily obtin = =. Substituting these esults bck to the Eq. yields: ξ + µ + ζ =, ξ + µ + ζ = Solving these two equtions simultneously yields Eq. 7: U U U = = = ξ η ζ ( ξ = µ, ζ = ± (7 Equilibium Points in the plne η=: The utho As expected fom the symmety of the poblem, we expects new set of equilibium points in the ξξ plne. obtined the loctions of the new equilibium points. It Now we cn enme the tingul Lgnge looks simil to Lgnge tingul equilibium points equilibium points L, L 5 to become L ξη, L ξη 5 this ( ξ, η = µ, becuse Lgnge found them in ξη. In this study the, µ, but except the plne utho found new set of equilibium points in the ξξ in which they lie: i.e., in the plne η = Eq. 8: plne. This new set will be clled in the futue woks Fwzy ξξ-tingul equilibium points L ξζ, L ξζ 5. To ( ξ, ζ = µ,, µ, (8 clculte the loctions of these new equilibium points, 79
Equilibium points in the plne pllel to ξ = plne: We conside test pticle t plne pllel to the plne ξ = (except t plnes ξ = -µ nd ξ = -µ becuse we did not find ny moe equilibium points thee. Accoding to these constints we hve Eq. 9: ( ξ + µ = + η + ζ = ( ξ + µ + η + ζ (9 = ξ + η + ζ Thus we cn wite: ( ( µ ξ + µ + η + ζ +µ ξ + µ + η + ζ = ξ + η + ζ µ µ Substituting into (: U = ( µ + + µ + + µ ( µ The system ( cn be witten s Eq. : ( ξ + µ U U i = = ( µ ξ i= i ξ ξ + µ +µ = U U η = = ( µ η i= η η +µ = U U i ζ = = ( µ ζ i= i ζ ζ +µ = Am. J. Applied Sci., 9 (9: 78-8, ( ( ξ + µ + η + ζ =, ( ( ξ + µ + η + ζ = Solving these two equtions simultneously fo ξ yields ξ = µ, substituting bck into Eq. 8 we get η + ζ = ζ = ± η povided tht Stbility of fwzy equilibium cicle: We e going to U U study the stbility of Fwzy equilibium cicle. At ech This system is veified when =, = Eq. step we will exmine ou solution using the well known esults in RTBP. To exmine the stbility, n infinitesiml body would be displced little fom the Fwzy equilibium cicle. If the esultnt motion of the ( µ =, µ = ( pticle is pid deptue fom the vicinity of this cicle we cn cll such cicle n unstble cicle, if howeve the pticle meely oscilltes bout the fom which one cn esily obtin = =. Substituting equilibium points of its cicumfeence, it is sid to be these esults bck to the Eq. 9 yields Eq. : stble positions. If we hve mixed sitution then we 8 η η. Theefoe we hve infinite numbe of solutions (equilibium points, thei coodintes e given by Eq. : ξ, η, ζ = µ, η, ζ = ± η ( Remk : As is cle fom the solution set we hve obtined infinte numbe of equilibium points which lie on cicumfeence of cicle centeed t µ,, nd its cicumfeence psses though the Lgnge nd Fwzy tingul equilibium points L ξη, L ξη 5, L ξζ ξζ, L 5. This cicle will be clled in the futue woks Fwzy equilibium cicle. i.e., The Lgnge nd Fwzy tingul equilibium points is subset of Fwzy equilibium cicle. Specil cses: To check ou solutions, setting η = ± in ( yields diectly the fmili Lgngin tingul equilibium points L ξη, L ξη 5 on espective t ( ξ, η = µ, ±. Also if we set η = Fwzy tingul equilibium points follow diectly s L ξζ, L ξζ 5 on espective t ξ, ζ = µ, ±.
Am. J. Applied Sci., 9 (9: 78-8, will detemine the stble points nd the unstble ones. To exmine the stbility of the obits in the vicinity of Fwzy equilibium cicle the equtions of motion e lineized ound the cicumfeence of the equilibium cicle with coodintes Eq. :... ξ = η = ζ = ξ = η = ζ = U ξ n η = ξ ξ= ξ U η + n ξ = η η= η U.. ζ = ζ ζ = ζ ( the subscipt indictes evlution fo ξ = ξ, η= η nd ξ = ξ. If Eq. e now evluted t ξ = ξ +ξ, ξ = ξ +ξ nd η= η +η, one cn get Eq. 5: ( ξξ ( ξη ( ξζ ξ n η = U ξ + U η + U ζ +... n = U U U... ( ξη ( ηη ( ηζ η + ξ ξ + η + ζ + ( ζξ ( ζη ( ζζ.. ζ = U ξ + U η + U ζ +... (5 We cn ewite Eq. 5 in mtix nottion s Eq. 6: Theefoe we get the chcteistic eqution, with degee n in λ with possible complex oots. Afte getting the eigenvlues λ s substitute them into (AλIX= nd solve fo the components X. The esulting equtions e coupled (diffeent components e ppeed in the sme eqution. Using the simility tnsfomtion Y = BX we cn tnsfom this coupled system to uncoupled one s: Y = BX X = B Y X = B Y B Y & = AB Y Y & = BAB Y whee, BAB - is digonl mtix with the eigenvlues of A on the digonl, the mtix B is constucted fom the n eigenvlues the mtix A. Now The coupled system (7 is tnsfomed to uncoupled Eq. 9: Y & = BAB Y = λ Y i i i i & & (9 The solutions of the tnsfomed system e esily found s Eq. : Y = c exp ( λ t i i i ( whee c i e n constnts of integtion. We must now tnsfom bck to the oiginl vibles in components fom s Eq. nd : ξ& ξ η& η ζ = && ξ U ξξ U ξη U ξζ ξ& η&& Uξη Uηη Uηζ η& Uζξ Uζη Uζζ && ζ (6 The point is switching fom the solution of simultneous second ode diffeentil equtions to the solution of system of fist ode diffeentil equtions. X ξ, η, ζ, ξ&, η& Let us denote to the stte vecto by ( ccodingly X & epesents ny of Eq. 6 becomes Eq. 7:........ ξ, η, ξ, η, ζ, thus X & = AX (7 To solve this system we compute fist the eigenvlues of the coefficient mtix A s follows Eq. 8: ( A λi X = det ( A λ I = (8 X = B Y = B c exp ( λ t i i i i ξ& λ ξ η& λ η λ ζ = && ξ U ξξ U ξη U ξζ λ ξ& η&& Uξη Uηη Uηζ λ η& Uζξ Uζη Uζζ λ && ζ λ λ λ det ( A λ I = = Uξξ Uξη Uξζ λ Uξη Uηη Uηζ λ Uζξ Uζη Uζζ λ ( ( which e line diffeentil equtions with constnt coefficients so long s only fist ode tems e etined. Let solution of the poblem be Eq. : 8
Am. J. Applied Sci., 9 (9: 78-8, ( t ( t ( t ξ = A exp λ η = Bexp λ ( ζ = Cexp λ whee, A, B, C nd λ e constnts. To find the expessions fo A, B nd C Eq. 5 cn be ewitten, using the suggested solution, s Eq. : ( λ ξξ ( λ + ξη ( ξζ ( ξη ( ηη ( ηζ ( ζξ ( ζη ( ζζ U A U B U C = λ U A + λ U B U C = ( U A U B + λ U C = which cn be witten in mtix nottion s Eq. 5: λ ξξ ( λ + ξη ( ξζ ( λ ξη ( λ ηη ( ηζ ( Uζξ ( Uζη ( λ Uζζ U U U A U U U B = C This system hs nontivil solution if Eq. 6: λ Uξξ ( λ + Uξη ( Uξζ ( λ ξη ( λ ηη ( ηζ ( Uζξ ( Uζη ( λ Uζζ U U U = expnding the deteminnt yields Eq. 7: ( ( ( ξξ ηη ζζ = λ + U U U λ 6 + U U + U U + U U ξξ ηη ξξ ζζ ηη ζζ U U U U λ ζζ ξη ξζ ηζ + U U U U U U ξξ ηη ζζ ξη ξζ ηζ + + + UξξUηζ UξηUζζ UξζUηη (5 (6 (7 whee σ is the oot of the chcteistic deteminnt nd U αβ e the second ptil deivtives with espect to the vibles mentioned in the subscipts nd evluted t the equilibium points. Evluting the ptil deivtives included in Eq. 7. Now computing the coefficients of λ ' s, then the chcteistic Eq. 7 cn be witten in the fom Eq. 8: ( 9 ( λ + λ + µ µ η + ζ ζ λ 6 + ξ η ζ 8 = (8 (setting ζ =, η = we ge Eq. 9t: 6 7 λ + λ + µ ( µ λ = 7 ( = λ + λ + µ µ which hs the fou solutions: λ,,, = ± ± 7µ 7µ +. (9 The only oscilltoy stble solution when λ is el nd negtive. In this cse two puely imginy oots λ = ± i λ exist. This cse is veified when: 7µ 7µ + i.e., when the mss tio becomes the vey well known esult µ =.85. Now etun bck to chcteistic Eq. 7. It hs the solutions Eq. : ( λ = ± 9µ µ η + ζ ζ ( 9 ( ± µ µ η + ζ ζ 8 µ η ζ η-dependent Stbility: Setting: Yields Eq. :, ζ = η, ξ = µ 7 λ ± µ ( µ + η = 9 7 ( ± µ µ + η 9 7 µ η η ( ( To check t this step let us etun bck to the stbility investigtion in the cse of Lgnge's tingul points: 7 quntity η 9, only when: 8 The quntity µ ( µ is positive definite nd the
Am. J. Applied Sci., 9 (9: 78-8, η, U, The quntity inside the inne sque oot cn be checked fo the negtive vlues s follows; setting: 7 F = 9 7 ( η µ ( µ + η ( µ ( η η 7 57 85 η ± + µ + µ 56 96 7 89 9 5 µ µ µ 687 769 599 99 ± + 85µ µ 5 5 975 + µ µ + 599µ ( Expnding F (η yields Eq. : 67 79 79 6 8 6 F η = µ µ µ + µ + 8 ( 7 7 ( 5 + + µ µ η + µ+ µ η ( which cn be witten s F ( η = + η + η, whee,, e the coefficients of η ' s in Eq.. Completing the sque nd seching fo the negtive vlues yields: F η = + η + < η ± ± ξ-dependent stbility: Recll Eq. nd setting = η ζ, ξ = µ yields: 7 λ = ± µ ( µ ζ 7 ( ± µ µ ζ 7 µ ζ ζ 7 The quntity µ ( µ ζ ζ µ ( µ, U, µ ( µ is positive when: ( 7 7 ( 5 µ + µ + µ µ η ± ± ( 5 µ + µ ( 7 7 ( 5 + µ µ µ + µ 67 79 µ + µ µ 6 8 79 + µ + 8 6 The quntity inside the inne sque oot cn be checked fo the negtive vlues s follows; setting: 7 F ( ζ = µ ( µ ζ 7 µ ζ ζ Expnding F( ζ yields Eq. : 79 F ( 6 6 ξ = µ µ + µ + µ µ ξ + ξ ( which cn be witten s F ( ζ = + ζ + ζ, whee 5 Expnding nd etining the tems up to O ( µ the, <, e the coefficients of ζ ' s in Eq.. Completing the sque nd seching fo the negtive bove eqution cn be ewitten in the fom Eq. : vlues yields: 8
Thus Eq. 5: F ζ = + ζ + < ζ ± ± ζ ± 8± µ µ µ ( 6 + 8µ + µ 56 Am. J. Applied Sci., 9 (9: 78-8, (5 As is cle fom Eq. nd 5 the genel solution fo the components of the position nd velocity ound the equilibium points of on the cicumfeence of the cicle involve line combintions, linely independent tems, of exp( + i λ t nd exp( i λ t tht leds by Eule identity to sines nd cosines. Thus we obtined oscilltoy stble solutions when η o ξ stisfies the citi deived in ( nd (5. ACKNOWLEDGMENTS The utho is deeply indebted nd thnkful fo the denship of the scientific esech fo his helpful nd fo distinct tem of the employees t Tibh univesity, Al-Mdinh Al-Munwwh, K.S.A. This esech study ws suppoted by gnt No. (6/. REFERENCES Bumbeg, V.G., 97. Reltivistic celestil mechnics. Nuk, Moscow. Depit, A. nd A. Depit-Btholome, 967. Stbility of the tingul lgngin points. Aston. J., 7: 7-79. Douskos, C.N.,. Equilibium points of the esticted thee-body poblem with equl polte nd diting pimies nd thei stbility. Astophys Spce Sci., : 79-87. DOI:.7/s59--58-7 Kum, S. nd B. Ishw,. Loction of colline equilibium points in the genelised photogvittionl elliptic esticted thee body poblem. Int. J. Eng., Sci. Technol., : 57-6. 8 Leontovich, A.M., 96. On the stbility of the esticted poblem of thee bodies. Soviet mth. Dokl., : 5-8. Mkeev, A.P., 969. On the stbility of the tingul libtion points in the cicul bounded thee-body poblem. J. Applied Mth. Mech., : 5-. DOI:.6/-898(6997-8 Nyn, A. nd C. Rmesh, 8. Stbility of tingul points in the genelized esticted thee body poblem. J. Mod. Ex-B, Fnce. Nyn, A. nd C.R. Rmesh,. Stbility of tingul equilibium points in ellipticl esticted thee body poblem unde the effects of photogvittionl nd oblteness of pimies. Int. J. Pue Applied Mth., 7: 75-75. Shnkn, S., J.P. Shm nd B. Ishw,. Equilibium points in the genelised photogvittionl non-pln esticted thee body poblem. Int. J. Eng., Sci. Technol., : 6-67. Shm, R.K. nd R.P.V. Subb, 986. On finite peiodic obits ound the equiltel solutions of the pln esticted thee-body poblem. Poceedings of the Intentionl Wokshop Spce Dynmics nd Celestil Mechnics, Nov. -6, Dodecht, D. Reidel Publishing Co., Delhi, Indi, pp: 7-85. Singh, J.,. Nonline stbility in the esticted thee-body poblem with oblte nd vible mss. Astophys Spce Sci., : 6-69. DOI:.7/s59--57-y Subb, R.P.V. nd R.K. Shm, 988. Oblteness effect on finite peiodic obits t L. Poceedings of the 9th IAF, Intentionl Astonuticl Congess, Oct. 8-5, Bngloe, Indi, pp: 6. Subb, R.P.V. nd R.K. Shm, 99, Stbility of L in the Resticted Thee-Body Poblem with Oblteness. st Edn., Vikm Sbhi Spce Cente, Indi, pp:. Subb, R.P.V. nd R.K. Shm, 997. Effect of oblteness on the non-line stbility of in the esticted thee-body poblem. Celest. Mech. Dyn. Ast. 65: 9-. Szebehely, V.G., 967. Theoy of Obits in the Resticted Poblem of Thee Bodies. Acdemic Pess, New Yok, ISBN-: 6865, pp: 668. Szebehely, V.G., 979. Instbilities in Dynmicl Systems: Applictions to Celestil Mechnics. st Edn., D. Reidel Pub. Co., Dodecht, ISBN: 97797, pp:.