Computation of the Locations of the Libration Points in the Relativistic Restricted Three-Body Problem

Transcription

1 Ameican Jounal of Applied Sciences 9 (5): , 0 ISSN Science Publications Computation of the Locations of the Libation Points in the Relativistic Resticted Thee-Body Poblem, Abd El-Ba, S.E. and,4 F.A. Abd El-Salam Depatment of Applied Math., Faculty of Applied Science, Taibah Univesity, Al-Madeenah Al-Munawwaah, Saudi Aabia Depatment of Mathematics, Faculty of Science, Tanta Univesity, Tanta, Egypt Depatment of Mathematics, Faculty of Science, Taibah Univesity, Al-Madeenah Al-Munawwaah, Saudi Aabia. 4 Depatment of Astonomy, Faculty of Science, Caio Univesity, Caio, Egypt Abstact: Poblem statement: In this study, the effects of Relativistic Resticted thee-body Poblem (in bief RRTBP) on the equilibium points of both tiangula and collinea is consideed. The appoximate locations of the collinea and tiangula points ae detemined. Seies expansions ae used to develope in µ and /c as small paametes. To check the validity of ou solution, when ignoing /c tems we get diectly the classical esults of the esticted thee-body poblem. Conclusion/Recommendations: A MATHEMATICA pogam is constucted to give a numeical application of the elativistic petubations in the locations of the equilibium points of the thee body poblem. Key wods: Relativistic esticted thee body poblem, location of the libation points, classical limit INTRODUCTION The thee-body poblem consides thee mutually inteacting masses m, m and m. In the esticted thee-body poblem, m is taken to be small enough so that it does not influence the motion of m and m, which ae assumed to be in cicula obits about thei cente of mass. The obits of the thee masses ae futhe assumed to all lie in a common plane. If m and m ae in elliptical instead of cicula obits, the poblem is vaiously known as the elliptic esticted poblem o pseudo esticted poblem Szebehely (967). The histoy of the esticted thee-body poblem began with Eule and Lagange continues with Jacobi (86), Hill (878), Poincae (957) and Bikhoff (95). Eule and Couvoisie (980) fist intoduced a synodic (otating) coodinate system, the use of which led to an integal of the equations of motion, known today as the Jacobian integal. Eule himself did not discove the Jacobi integal which was fist given by Jacobi (86) who, as Wintne emaks, ediscoveed the synodic system. The actual situation is somewhat complex since Jacobi published his integal in a sideeal (fixed) system in which its significance is definitely less than in the synodic system. Many authos hope to investigate the elativistic effects in this poblem. But unfotunately, the Einstein field equations ae nonlinea and theefoe cannot in equiements of time independence and spatial isotopy we ae able to find one useful exact solution, the Schwazschild metic, but we cannot actually make use of the full content of this solution, because in fact the sola system is not static and isotopic. Indeed, the Newtonian effects of the planet s gavitational fields ae an ode of magnitude geate than the fist coections due to geneal elativity and completely swamp the highe coections that ae in pinciple povided by the exact Schwazschild solution. It is woth noting to highlight some impotant aticles in this field. Computed the post-newtonian deviations of the tiangula Lagangian points fom thei classical positions in a fixed fame of efeence fo the fist time, but without explicitly stating the equations of motion. Teated the elativistic (RTBP) in otating coodinates. He deived the Lagangian of the system and the deviations of the tiangula points as well. Weinbeg (97) calculated the components of the metic tenso by using the post-newtonian appoximation in ode to obtain the (RTBP) poblem equations of motion. Soffel (989) obtained the angula fequency ω of the otating fame fo the elativistic two-body poblem. Bumbeg (97; 99) studied the poblem in moe details and collected most of the impotant esults on elativistic celestial mechanics. In this study, the appoximate positions of the geneal be solved exactly. By imposing the symmety collinea and tiangula points ae detemined using the Coesponding Autho: Abd El-Ba, S.E., Depatment of Mathematic, Faculty of Applied Science, Taiba Univesity, Al-Madeenah, K.S.A, 659

2 equations of motion of the elativistic esticted theebody poblem. The fomulas ae expanded in tems of µ and as small paametes c Equations of motion: The equations of motion of the infinitesimal mass in the (RRTBP) in a synodic fame of efeence (ξ, η), in which the pimay coodinates on the ξ -axis (-µ, 0), (-µ, 0) ae kept fixed and the oigin at the cente of mass, is given by Bumbeg (99), Fom Bhatnaga and Hallan (997) Eq. : Am. J. Applied Sci., 9 (5): , 0... U d U ξ n η= dt... U d U η + n ξ = η dt η () whee, U is the potential-like function of the (RRTBP), which can be witten as composed of two components, namely the potential of the classical (RTBP) potential U c and the elativistic coection U Eq. : U = U c + U () whee, U c and U ae given by Eq. : U c µ µ = + + () And Eq. 4:. U = ( µ ( µ ) ) + (( ξ + η) c 8c. µ µ c + ( η ξ ) ) µ µ (( ξ+ η ) + ( η ξ) ) + c. + ( µ 7ξ 8 η ) +η c µ µ + µ µ with: (4) Fig. : The five Lagangian points with (Sun-Eath system; as an example) Location of the libation points: Fom the equations of motion (), it is appaent that an equilibium solution exists elative to the otating fame when the patial deivatives of the pseudopotential functions (U ξ, U η, U ζ ) ae all zeo, i.e., U = const. These points coespond to the positions in the otating fame at which the gavitational and the centifugal foces associated with the otation of the synodic fame all cancel, with the esult that a paticle located at one of these points appeas stationay in the synodic fame. Thee ae five equilibium points in the cicula (RTBP), also known as Lagange points o libation points. Thee of them (the collinea points) lie along the ξ-axis: one inteio point between the two pimaies and one point on the fa side of each pimay with espect to the baycente. The othe two libation points (the tiangula points) ae each located at the apex of an equilateal tiangle fomed with the pimaies. One of the most commonly used nomenclatues (and the one that we will use hee) defines the inteio points as L, the point exteio to the small pimay (the planet) as L and the point L as exteio to the lage pimay (the Sun), see Fig.. The tiangula points ae designated L 4 and L 5, with L 4 moving ahead of the ξ-axis and L 5 tailing ξ-axis, along the obit of the small pimay as the synodic fame otates unifomly elative to the inetial fame (as shown in Fig. ) Remak: All five libation points lie in the ξ-η plane, i.e., in the plane of motion The libation points ae obtained fom equations of motion afte setting ɺɺ ξ = η ɺɺ = ɺɺ ζ = 0. These points epesent paticula solutions of equations of motion Eq. 5: n = + ( µ ( µ ) ), = ( ξ + η ) c, = ξ + µ + η = ξ + µ + η U Uc U U Uc U = + = 0, = + = 0 (5) η η η The explicit fomulas ae (Remembeing that U = U c + U ) Eq. 6: 660

3 Am. J. Applied Sci., 9 (5): , 0 U ( µ )( ξ + µ ) µ ( ξ + µ ) = ξ µ µ + µ µ ξ + + c ( µ )( ξ + µ ) µ ( ξ + µ ) ( + + η + ξ ) ξ ( µ )( ξ + µ ) µ ( ξ + µ ) + η + ξ µ µ + + µ ( µ ) ( ξ + µ ) ( ξ + µ ) ( ξ + µ ) + + ( + µ + 7 ξ) µ ( ξ + µ ) ( µ )( ξ + µ ) η + = And Eq. 7: U ( µ ) µ =η η η+ η c µ µ ( µ ) µ ( µ µ ) η η ( µ ) µ + ( ξ + η ) η ( ξ + η ) + η ( µ ) + ( + ) η + η + µ ( µ ) η ( µ ) µ + ( + µ + 7 ξ ) + + η 5 5 µ µ + + = 0 Location of collinea libation points: Location of L. (6) (7) The collinea points must, by definition, have ξ = η = 0, since / c << and the solution of the classical (RTBP) is (Fig. ) Eq. 8: Fig. : Shows the location of L Using (8) Eq. 6, can be witten explicitly in tems of and as Eq. 9: U µ µ = µ µ µ µ µ + + c + ( µ µ )( µ ) + ( µ ) µ µ µ µ ( µ ) + + ( µ ) µ ( µ ) + + ( + µ + 7( µ )) 7 = 0 (9) Then it may be easonable in ou case to assume that position of the equilibium points L ae the same as given by classical (RTBP) but petubed due to the inclusion of the elativistic coection by quantities ε O / c Eq. 0: ( = a +ε, = b - ε, a = - b (0) whee, a and b ae unpetubed positions of and espectively and b is given afte some successive appoximation by: α α 4 µ b = α + α + α ; α = ( µ ) + =, = ξ + µ, = µ ξ, = = (8) Substituting fom Eq. 0 into Eq. 9 and etaining the tems up to the fist ode in the small paamete ε, we get: 66

4 U µ ε = µ b + ε ( b ) b µ µ µ µ ε b µ µ b c + + ( b ) b ( b ) µ µ + ( µ b ) ( b ) b µ µ ( µ b ) ( b ) b µ µ + + ( µ b ) ( b ) b µ ( µ ) 7 ( b ( b ) ) b + + (6 7b 4 µ ) = 0 ( b ) b Setting, ( µ ) µ + + d, e, = = ( b ) b b = f, = g, = h b ( b ) b Eq. - can be solved fo ε to yield: { ( ε = d + b +µ+ ( µ ) h µ f ( µ µ ) ( µ b ) c + µ +µ µ µ ( g e )( h f ) + ( µ b ) ( ( µ )h µ f )( µ b ) ( ) h 7( g e ) + ( µ )g +µ e ( µ b ) µ ( µ ) + h + f (6 7b 4 µ ) Am. J. Applied Sci., 9 (5): , 0 () () Expanding b, d, e, f, g and h in ode of µ µ etaining tems up to ode. Then substituting back into Eq.s 0 and 8 we get the location fo the fist libation point ξ o, L Fig. : Shows the Location of L ξ 0,L = { } 5 µ µ 6 µ 9 58 µ µ 4 µ µ 400 µ 56 µ µ 5 µ 4 µ 45 µ { + + c µ 695 µ 985 µ µ µ } With a and b ae the unpetubed positions of and espectively, whee b is by: 66 7 () In this equation tems that ae not factoed by / c epesent the location of ξ o, L in the classical RTBP while the tems that factoed by / c epesent the coection due to the inclusion of the elativistic tems. Location of L : The L point locates outside the small massive pimay of mass µ. We now dive an appoximate location fo this point. At the L point we have (see Fig. ) Eq. 4: =, = ξ + µ, = ξ + µ, = = (4) The pocedue is simila to Eq. 0 with little modification accoding to the location of ξ o, L as: = a + ε, = b + ε, a = + b

5 Am. J. Applied Sci., 9 (5): , 0 α α b = α + α + α, µ α = ( µ ) As is done in the calculation of the location of ξ o, L we can similaly calculate the location of ξ o, L as Eq. 5: µ µ 8 µ { ξ 0,L = } { c µ 87 µ 9545 µ ( ) } (5) Location of L : The L point lies on the negative ξ- axis. We now deive an appoximate location fo this point. At the L point we have (Fig. 4) Eq. 6: =, = ξ µ, = µ ξ, = = (6) The pocedue is simila to Eq. 0 with little modification accoding to the location of ξ o, L as: = a + ε, = b + ε, a = b whee a and b ae the unpetubed values of and espectively and b is given afte some successive appoximation by: b = µ + µ + µ As is done in the calculation of the location of ξ o, L, we can similaly calculate the location of ξ o, L as Eq. 7: Fig. 4: Shows the Location of L Location of the Tiangula Points L4,5: Since / c << and the solution of the classical esticted thee-body poblem is = =, then it may be easonable in ou case to assume that the positions of the equilibium points L 4,5 ae the same as given by classical esticted thee-body poblem but petubed due to the inclusion of the elativistic coection by quantities (, O( / c ) ε )Eq. 8: = ( +ε ), = ( +ε ) (8) Substituting in the second set of (7) and (8) and solving fo ξ and η up to the fist ode in the involved small quantities ε and ε, we get: µ ξ = ( ε ε + ) η = ± + ( ε + ε ). (9) Substituting the values of,, ξ and η into equations: U Uc U U Uc U = + = 0, = + = 0 η η η Evaluating the included patial deivatives and etaining the tems up to the fist ode in the small paametes ε, ε and also the fist ode tems in the elativistic coection, we obtained Eq. 0: ξ 0,L = { µ µ µ µ...} { µ + µ 98 c µ + µ +...} ( µ ) ε µε µ ( µ ) 8c (7) ( µ ) = 0,( µ ) ε +µε 7 µ ( µ ) = 0. 8c 66 (0)

6 Which epesent two simultaneous equations in ε and ε. Thei solutions ae being Eq. : ε = µ ( + µ ) 8c () ε = ( µ )(5 µ ) 8c Substituting the values of ε and ε into Eq. 9, yields the coodinates of the tiangula points Eq. : Am. J. Applied Sci., 9 (5): , 0 ( µ ) 5 ξ 0,L = ( + ) 4,5 4c η 0,L =± µ µ + c ( (6 6 5)) 4,5 () The classical limit: To check ou solution, when ignoing / c tems we get diectly the classical esults of the esticted thee-body poblem as: µ µ µ ξ 0,L = µ 4 µ µ µ 400 µ 56 µ Fig. 5: Location of the Equilibium point L vesus the mass atioµ. µ µ 8 µ ξ 0,L = { 7 50 µ 4 µ 40 µ 554 µ µ 665 µ } ξ 0,L = µ µ µ µ ( µ ) ξ 0,L =, η 4,5 0,L =± 4,5 Fig. 6: Location of the Equilibium point L vesus the mass atioµ. Numeical esults and concluding emaks: We have In Fig. 5-8, [L ic, i =,, ] denotes fo the position of used the above mentioned analysis and the explicit the equilibium points without the elativistic fomulas obtained to design a compute pogam using contibution, i.e within the classical poblem of the MATHEMATICA softwae. We have plotted the esticted thee bodies, while [L ir, i =,, ] denotes locations of the equilibium points L, L, L, vesus the fo the position of the equilibium points including the whole ange of the mass atio µ [0, 0. 5]. elativistic contibution. 664

7 Fig. 7: Location of the Equilibium point L vesus the mass atioµ Fig. 8: Location of the Equilibium points L, L, L vesus the mass atioµ. In Fig. 5 the diffeence L ir -L c seems to be constant beyond µ In Fig. 6 the diffeence L R - L c inceases with inceasing the mass atioµ. In Fig. 7 the diffeence L R - L c inceases with inceasing the mass atio µ within the domain 0 µ 0.5 and deceases with inceasing µ within the domain 0.5 µ 0.5. Fig. 8 epesents an assembly plot of the all collinea equilibium points. Am. J. Applied Sci., 9 (5): , We may see that, in most of these cases, the positions of L and L ae much moe affected by the influence of the elativistic tems than that of the equilibium point, L ACKNOWLEDGMENT The eseaches ae deeply indebted to the Pofesso D. M. K. Ahmed, the pofesso of space dynamics at Caio univesity, faculty of Science, depatment of Astonomy fo his valuable discussions and citical comments and ideas that help us to finalize this study. This eseach wok was suppoted by a gant No (64/4) fom the deanship of the scientific eseach at Taibah univesity, Al-Madinah Al-Munawwaah, K.S.A. REFERENCES Bhatnaga, K.B. and P.P. Hallan, 997. Existence and stability of L4,5 in the elativistic esticted theebody poblem. Celest. Mech., 69: 7-8. DOI: 0.0/A: Bikhoff, G.D., 95. The esticted poblem of thee bodies. end. Cicolo. Mat. Palemo, 9: DOI: 0.007/BF00598 Bumbeg, V.A., 97. Relativistic celestial mechanics. Pess Sci., Moscow, Nauka Bumbeg, V.A., 99. Essential Relativistic Celestial Mechanics. st Edn., Hilge, Bistol, ISBN-0: pp: 6. Eule, L. and L. Couvoisie, 980. Theoia Motuum Lunae Nova Methodo Petactata: Opea Mechanica Et Astonomica. Couvoisie, L. (Ed.). pp:. Hill, G.W. and N.N.Y. Tunpike, 878. Res. Luna Theoy, Ch. I. Am. J. Math., : 5-6. Jacobi, C.G., 86. Su le Movement d'un point et suun cas Paticulie du Pobleme des tios Cops, Compte Rendus de l Acad. Des. Sci., Pies, : Poincae, H., 957. Les Methodes Nouvelles De La Mecanique Celeste. st Edn., Dove, New Yok, pp: 44. Soffel, M.H., 989. Relativity in astomety, celestial mechanics and geodesy. st Edn., Spinge-Velag, Belin; New Yok, ISBN-0: pp: 08. Szebehely, V.G., 967. Theoy of obits: The esticted poblem of Thee-Bodies. st Edn., Academic Pess Inc, New Yok, pp: 668. Weinbeg, S., 97. Gavitation and Cosmology Pinciples and Applications of the Geneal Theoy of Relativity. st Edn., Wiley, New Yok, ISBN-0: pp: 657.