Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem F.. bd El-Salam, abah Unversty, Department of Mathematcs, Faculty of Scence, l-madnah l-munawwarah, Saud raba Caro Unversty, Faculty of Scence, Department of stronomy, Caro, 6, Egypt Emal: f.a.abdelsalam@gmal.com bstract In the present work the three dmensonal restrcted three body problem (n bref DRBP s formulated. he relatve equlbra are obtaned. he locaton of Fawzy equlbrum crcle s dscovered (n bref FEC. he Hamltonan near any pont on the crcumference of FEC s constructed. he nfntesmal orbts near FEC are derved usng an approach developed by Delva and Hanslmeer. Implct formulas for the poston and momentum vectors for n th order and explct formulas for the the same vectors for the frst, second and thrd orders are obtaned. Keywords: Restrcted hree Body Problem, Fawzy Equlbrum Crcle. Introducton he three-body problem s a very rch dynamcal system n mathematcal ntrcacy and practcal applcablty. he restrcted three-body problem nterested n the moton of a partcle of neglgble mass n the presence of two massve bodes. he subject of the perodc solutons of the restrcted problem of three bodes acqured great mportance and nterest snce the last decades of the 0 th century. he man reason s the need for space msson orbts n the vcnty of one of the colnear or trangular lbraton ponts. here are fve equlbrum ponts n the classcal three body problem, namely L, =,,,, 5, these ponts usually called Lagrangan or lbraton ponts. bd El-Salam (0 dscovered the so called Fawzy ξζ - trangular equlbrum ponts n the plane η = 0 ; the perpndcular plane to the plane of moton of the prmares. he author also dscovered the so called Fawzy equlbrum crcle n the three body problem. hs crcle passes through the Lagrange's and Fawzy trngular ponts. he lterature s rch and the works dealng wth the perodc orbts near the equlbrum ponts cannot be exhaustvely revewed However, t wll be benefcal to sketch some of these most mportant works. n early and very useful analyss of the behavuor of the bodes near lbraton ponts has been compled by a group of emnent scentsts n a work by Duncombe and Szebehely (966. Rchardson (980, Barden and Howell (998, Howell (00, Gomez et al. (998; 00, Selaru and Dmtrescu (995, Namoun and Murray (000, Corbera and Llbre (00, shraf Hamdy et al. (005. he motvaton of the present work s to study the the nfntesmal orbts near any pont on the crcumference of Fawzy equlbrum crcle (he recently dscovered equlbrum
Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem: F.. bd El-Salam crcle. smlar treatment has appeared n the work by shraf Hamdy et al. (005, but they has appled the Delva-Hanslmeer technque for the nfntesmal orbts near the Lagrange's equlbrum ponts. hree Dmensonal RBP Let r,r R beng the postons of the two massve bodes and respectvely, whch s the soluton of the two-body problem. Let the frst mass be µ ] 0,[, thus the second mass has µ and the gravtatonal constant s equal to. Let us denote the center of mass R = µ r + ( µ r one can easly prove that _ d R =0. ransformng to co-movng coordnates r = r R, =, and let r R denotes the poston of the test partcle, these yeld the equatons of motons n these new varables as; ( _ d r µ µ = rr rr rr rr ( t ths pont we start makng the followng assumptons: ( he prmares move n a crcular plannar orbt around ther center of masss wth constant angular velocty normalzed to wthout loss of generalty. ( he frame of reference s rotatng wth the rotaton matrx defned below. In ths coordnate system the prmares become statonary and le along x - axs. ( he test partcle moves n a plane perpndcular to the prmares'plane, wth coordnates x = ( x, x, x. herefore we set, _ ( t[ x x x ] =, r R _ ( t [ µ ] r = R 0 0, _ ( t [ µ ] r = R 0 0 n whch R ( t s the rotaton matrx, gven below, wth the followng propertes, cost snt 0 R ( t = snt cost 0 0 0 0 0 d R ( t dr ( t = R ( t, R ( t = J = 0 0 0 0 0 We can deduce the equatons of moton for x R : ( x [ ] x [ ] ( ( x [ ] x [ ] d x dx µ µ 0 0 µ µ 0 0 x + J = µ 0 0 µ 0 0 5
Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ Fnally, settng = d x p J + x we fnd that these are the Hamltonan equatons of moton on { [ ] [ ] } x µ µ R 6 / = 0 0, 0 0 wth Hamltonan µ µ ( p p ( ξp ηp η ξ = x 0 0 0 0 H =. ( [ µ ] x [ µ ] 6 where we have equpped R wth the canoncal symplectc form d ξ dp + dη dp,.e. the ξ η equatons of moton are gven by dx H dp H =, =. p x Relatve Equlbra he relatve equlbra are the the solutons of the Hamltonan vector feld gven by (. Let us wrte the gravtatonal potental energy functon as µ µ Ω( x = x [ 0 0] µ x [ µ 0 0] ( he equlbrum solutons of ( are obtaned settng all the partal dervatves of H equal to zero as; x x x x x x x ( x ( x H = p + x = 0 H = U = 0 p + p x x ( x H = p x = 0 H = U = 0 p + p x x H = px = 0 H = U = 0 p x x or equvalently ( x ( x ( x Ω Ω Ω = x, = x, =0 x x x ( where p at the equlbrum pont can easly be found once we solved ( for x at the equlbrum pont. Note that x solves ( f and only f x s a statonary pont of the functon U = x x ( x ( + Ω( x called the amended potental. Let us frst look for equlbrum ponts of the amended potental that le on the lne of syzygy (he lne on whch all the three bodes
Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem: F.. bd El-Salam smltanuously le at some specfed date,.e. for whch x =0. Note that automatcally satsfed n ths case snce Ω( x U ( x =0, =0 reduces to x x x =0 U x ( x =0 s d d µ µ U ( x,0 = x + + =0 + + µ µ dx dx x x (5 Clearly, ( lm ( lmu x,0, U x,0, x x µ (,0, lm (,0 lm U x U x x µ x so U ( x has at least one crtcal pont on each of the ntervals,0 ], µ [, ] µ, µ [, ] µ, [ o explore the concavty of the mentoned ntervals, we calculate the second dervatve as; µ µ,0 = >0 dx x x d ( U x + + + µ + µ (6 So U ( x,0 s convex on each of these ntervals and we conclude that there s exactly one crtcal pont n each of the ntervals. he three relatve equlbra on the lne of syzygy are called the Euleran equlbra. hey are denoted by L ; L and L, where L ], µ [ { 0}, L ] µ, µ [ { 0} and L ] µ, [ { 0}. Now we shall look for new equlbrum ponts that do not le on the lne of syzyg. Let us use ξ = ( x + µ + x + x, ( η = x µ x x, ζ = x + x + x + + + as coordnates n each of the half-planes x > 0 and x < 0. hen U can be wrtten as (7 U = η ξ ( µ η + + µ ξ + µ ( µ (8 U U U t the equlbrum ponts = = =0 ξ η η 7
Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ U = µ ξ = 0 ξ =, ξ ξ U = ( µ η =0 η = η η Substtutng ξ =, η = back to the frst two equatons of (7 yelds ( µ x + + x + x = x + + x + x = ( µ (9 We can solve ths system of two equatons n three varables yelds two varables namely; x whch wll have a fxed value, and x whch wll have a wde range as we shall see below. he thrd varable x can be expreesed n terms x. Subtractng the two equatons of (9 yelds x + µ x + µ = x + µ = x + µ x = µ ( ( 0 ( ( Substtutng provded that x = µ back nto any equaton of (7 we get x + x = x = ± x x x. herefore we have nfnte number of solutons (ncludng Lagrange trangular equlbrum ponts and Fawzy trangular equlbrum ponts, ther coordnates are gven by ( x, x, x = µ, x, x = ± x (0 ransformed Hamltonan Near FEC t ths pont we are nterested n the nfntesmal orbts near any pont on the crcumference of FEC. Movng the orgn to any pont on FEC and denotng to the new X, X, X, P, P, P by the followng substtuton coordnates and momenta by ( X X X ξ = X + x, η = X + x, ζ = X + x, p = P X x, p = P ξ X + x, η
Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem: F.. bd El-Salam where µ. ( x, x, x =, x, x = ± x hese settngs yelds the transformed Hamltonan as; = H ( P + P + ( X P X P + X + x X x X X X X X 8 + + µ X + x X + x X x X X x X X µ µ 9 9 x x X X x x ( x ( 8 + + µ 8 µ µ µ ( Perturbaton pproach and Solutons We use an approach developed by Delva (98 and Hanslmeer (98 n whch the procedure can be performed wth a dfferental operator D. specal lnear operator, the Le operator, produces a Le seres. he convergence of the seres s the same as for aylor seres, snce the Le seres s only another form of the aylor seres whose terms are generated by the Le operator. We wll use ths Le seres form for two reasons. he frst s the requrement to buld up a perturbatve scheme at dfferent orders of the orbtal elements. he second s ts usefulness n treatng the non-autonomous system of dfferental equatons and non-canoncal systems. hs enables a rapd successve calculaton of the orbt. In adon we can change the stepsze easly (f necessary. hs s an mportant advantage for the treatment of the problems whch has a varable stepsze, e.g. for the mass change of the prmares. he formulas has an easy analytcal structure and may be programmed wthout dffculty and wthout mposng extra conons on the convergence. Snce any desred number of terms can be found by teraton, the seres can be contnued up to any satsfactory convergence reached. D X P X P ( ( X ( X Ξ dx Ξ dp X Ξ= + +, Ξ =X,,P, ( = X P X t th Usng Lebntz formula for the n dervatve of a product, namely n! ( ( = n m nm d m d gd h n n g z h z C, C = n n m n m m dz m =0 dz dz m!( n m! yelds the n th ( n applcaton of the Le operator denoted by D as; D Ξ d X Ξ d P Ξ Ξ ( X P X t n m n m nm m n ( n m X = Cn + m n m m n m + n = m=0 Let X X X P P P X X X H (,,,,, be the Hamltonan functon near any pont on the crcumference of Fawzy equlbrum crcle as gven by (, and usng the canoncal 9
Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ dx H dp equatons of moton =, X H = P X X to evaluate the dervatves d X nm, n m then we can reach to the solutons (coordnate and momentum vectors, X, P respectvely as; { } j ( t t D j ( n ( n ( e D D X + D X, X=X X=X 0 j=0 0 P=P0 j=0 X=X0, P=P0 j ( t t ( t t X= X = X = ( j! j! d nm P X n m j ( t t D j ( n ( n ( n ( e D { D P + D P + D P }, 0, X X X X=X0 P=P X=X P=P 0 j=0 0 j=0 X=X0, P=P0 j ( t t ( t t P = P = P = (5 j! j! Solutons at Dfferent Orders In ths secton we are gong to evaluate the solutons at dfferent orders. From the defnton ( n of operator D, gven n (, we get the followng expressons for the coeffcents. he Frst Order Soluton Settng n = we have the requred coeffcents n the equatons (, (5 to yeld the frst order soluton as; ( D X = P X, ( X he Second Order Soluton D X = P + X, X ( D P = P +, X X x µ x X + xx µ µ ( D P = P, X X x X x + xx + xxx µ ( D P = xx xxx ( x X X x µ + Settng n = we have the requred coeffcents n the equatons (, (5 to yeld the second order soluton as; ( 5 D X = P +, X x µ x X + x X µ µ ( D X = µ xx ( x X + xxx P x, X ( 5 5 D P = X x µ x X P + µ x X x x x X µ + + xxx + x x, µ
Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem: F.. bd El-Salam ( 5 D P = x µ ( P x P + µ x x x + X x X + µ x X +, µ µ ( D P = x X µ P + xxp xxx x X X X µ X X X he hrd Order Soluton Settng n = we have the requred coeffcents n the equatons (, (5 to yeld the second order soluton as; ( D X = + x xµ 6 P + X µ xx 6x + x µ x X + 6xxX x, ( D X = + 6µ x ( xx + µ xx µ xx + µ xp ( x P + (, X X µ ( 5 D P = x 6 X µ P + x + x x P X X µ 5 5 5 + 9µ x 9µ x + µ x x x + X 6 5 + x µ X x x x X + µ + µ 5 µ x x +, µ ( D P = 6 ( X x x x + P + x P X X µ µ x ( µ ( 6x + x µ x 5 X + 9x x x + x + X 8 µ 5 5 + x x 9 9 9 µ µ + x X + x + x + µ µ ( D P =6x x P 6x P x ( ( x 6x x 5 X X X µ + + X µ µ 8 + x x ( + x X + 9 µ x + 9x x X + x x x µ Now we can rewrte the soluton up to the thrd order as; X = P X = n= = n= = X { M X + N P + C, n, n X n} P { X + B P + C X, n, n X n } X=X0, P=P0 X=X0, P=P0 ( t t n! ( t t n! n n
Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ where M = = M M,, =η x + µ x 6µ x = x µ x + x x + x = x,, (, N == N B,, =x + x x, = = B, 0, 5 M =x 6µ x, M =, x x + N = = N M B B M N M N, 0, ( µ 5 = x x x + 5 =x x + x µ + x 6µ x = ( x µ + x =6x 5 =x x + µ x 6µ x =x + x x,,,,,, = ( 6 7 x x =x 6µ x + = 9x + 9x x x 6x x + =x µ + x,,, (, =x, 9 5 = x µ x x x + 9µ x + 9x x 8µ x x 5 9µ x + 9x x + 8µ x x x 8 5 B = 6x x + 6µ x + x µ x, B = 6x 6x x +, X C = x, µ X C =x µ,
Infntesmal Orbts Near ny Pont on Fawzy Equlbrum Crcle n the hree Dmensonal Restrcted hree Body Problem: F.. bd El-Salam C = µ x x, = x µ x µ x + 5 5 = x µ + x µ x x + 6µ x + µ x x x 6µ x + 8 5 5 5 9 9 5 = x x µ x x x + x + x + 8µ x + µ x + 9x x 8 8 5 8µ x 9µ x + 9µ x x 8µ x x 9µ x x + 6 P X C P X C P X, Concluson and Outlook hree dmensonal restrcted three three body problem s constructed. he relatve equlbra are obtaned. n equlbrum crcle s dscovered namely Fawzy Equlbrum crcle (n bref FEC. Infntesmal orbts near FEC usng an approach developed by Delva and Hanslmeer are derved. We obtaned mplct formulas for the poston and momentum vectors for n th order and explct formulas for the the same vectors for the frst, second and thrd orders. In forthcomng works we am to evaluate the perodc orbts near FEC n the ellptc and/or oblate restrcted three body problem. lso we hope to nvestgate the perturbed locaton and stablty of FEC n ths framework. he same ponts wll also be treated n the doman of the relatvestc restrcted three body problem. cknowledgments: he author s deeply ndebted and thankful to the dean of the scentfc research and hs helpful and dedcated team of employees at the abah unversty, l-madnah l-munawwarah, K.S.. hs research work was supported by a grant No. (600/. lso I wsh to express my deep grattude and very ndebtedness for the referees for all very frutful dscussons, constructve comments, remarks and suggestons on the manuscrpt. References shraf Hamdy, Mostafa, K. hmed, Mohmad Radwan and Fawzy hmed bd El-Salam (00 Ifntesmal orbts around the lbraton ponts n the ellptc restrcted three body problem. Earth, Moon and Planets, v. 9, pp. 6-7. bd El-Salam, F.. (0 Dscovery of an equlbrum crcle n the crcular restrcted three body problem. mercan Journal of ppled Scences, v. 9(9, pp. 78-8, do: 0.8/ajassp. 0.78.8. Barden, B.. and Howell, K. C. (998 Fundamental motons near collnear lbraton ponts and ther transtons. J. stronaut. Sc., v.6, pp. 6-78. Corbera, M., Llbre, J. (00 Perodc orbts of a collnear restrcted three body problem. Celest. Mech. Dyn. stron., v. 86(, 6-8. Duncombe, R. and Szebehely, V. (966 Methods of astrodynamcs and celestal mechancs. Progress n astronautcs and aeronautcs ; v. 7, cademc press, New York, xv, 6p. Delva, M. (98 Integraton of the ellptc restrcted three-body problem wth Le seres. Celest. Mech., v., pp.5-5. Gomez, G., Jorba,., Masdemont, J. and Smo, C. (998 Study of the transfer between halo orbts. cta stronaut., v., pp. 9-50. Gomez, G., Koon,W. S., Marsden, J. E., Masdemont, J. and Ross, S. D. (00 Connectng orbts and nvarant manfolds n the spatal restrcted three-body problem. Nonlnearty v. 7(5, pp. 57-606. Hanslmeer,. (98 pplcaton of Le seres to regularzed problem n celestal mechancs. Celest. Mech., v., pp. 5 -. Howell, K. C. (00 Famles of orbts n the vcnty of the collnear lbraton ponts. J. stronaut. Sc., v. 9(, pp. 07-5. Namoun, F. and Murray, C. D. (000 he effect of eccentrcty and nclnaton on the moton near the Lagrangan Ponts L, L 5. Celest. Mech. Dyn. stron., v. 76. pp.-8.
Earth Scence Inda, eissn: 097 850 Vol. 6 (I, January, 0, pp. - http://www.earthscencenda.nfo/ Rchardson, D. L. (980 nalytc Constructon of Perodc Orbts bout the Collnear Ponts. Celest. Mech., v., pp. -5. Selaru, D. and Cucu-Dumtrescu, C. (995 Infntesmal orbt around Lagrange ponts n the ellptc restrcted three body problem. Celest. Mech. Dyn. stron., v. 6(, pp. -6. (Submtted on: 8.7.0; ccepted on: 9..0