Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 Scholars Journal of Phscs Mathematcs Statstcs Sch. J. Phs. Math. Stat. 5; (B):69-75 Scholars Academc Scentfc Publshers (SAS Publshers) (An Internatonal Publsher for Academc Scentfc Resources) ISSN 9-856 (Prnt) ISSN 9-864 (Onlne) The Stablt of the Trangular Ponts of the Restrcted three Bod Problem when both the Prmares are Traal Rgd Bodes S. M. Elshabour M.R. Amn Department of Math. Facult of Scence An Shams Unverst Caro Egpt Departement of theoretcal phscs Natonal Research Centre Caro Egpt. *Correspondng Author: M.R. Amn Emal: mohammad.nrc@gmal.com Abstract: The locaton the stablt of the trangular ponts of the planar restrcted three bod problem have been dscussed when both the prmares are traal rgd bodes consderng the case of statonar rotatonal moton of the bgger prmar of the smaller prmar are respectvel. Kewords: restrcted three bod problem; trangular ponts; traal rgd bodes. INTRODUCTION The problem of stablt condtons of trangular lbraton ponts was assumed b Gascheau [] then b Routh[5]. In recent tmes man perturbng forces.e. oblateness radaton forces of the prmares Corols centrfugal forces etc. have been ncluded n the stud of the restrcted three bod problem. Bhatnagar Gupta[] show the estence of 6 statonar motons each correspondng to the constant values of the non-cclc generalzed coordnates thus dependng on the Euleran angles of both the bodes. Khanna Bhatnagar [] have studed the problem when the smaller prmar s a traal rgd bod. Also Sharma et.al.[5] have studed the problem when both the prmares are traal rgd bodes n the case of statonar rotatonal moton are small quanttes. In ths paper we consder the restrcted three bod problem when both the prmares are traal rgd bodes wth the statonar rotatonal moton of the bgger prmar of the smaller prmar. EQUATIONS OF MOTION We shall adopt the notaton termnolog of Szebehl [7]. As a consequence the dstance between the prmares does not change s taken equal to one; the sum of masses of the prmares s also taken one. The unt of tme s chosen so as to make the gravtatonal constant unt. Besdes ths the prncple aes of the prmares are orented to the snodc aes b Euler's angels. Snce the aes are supposed to rotate wth the same angular veloct as that of the rgd bodes the bodes are movng around ther center of mass wthout rotaton the Euler's angles reman constant throughout the moton. Usng dmensonless varables Avalable Onlne: http://saspjournals.com/sjpms 69
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 Fg-: Left: the crcular restrcted three bod problem n the snodcal reference sstem wth a dmensonal unts. Rght: the fve equlbrum ponts assocated wth the problem. the equatons of moton of the nfntesmal mass m n a snodc coordnate sstem are... n... n n r r I I I I r r m r I ' I ' I ' I ' mr () () r r () Here s the rato of mass of the smaller prmar to the total mass of prmares.e. m wth m m beng the masses of the prmares. m m I I I are the prncpal moments of nerta of the traal rgd bod of mass m at ts center of mass wth a b c as ts aes. I s the moment of nerta about a lne jonng the center of the rgd bod of mass m the nfntesmal bod of mass m s gven b I I l I m I n ' ' ' l m n are the drectonal cosnes of the lne respect to ts prncpal aes. ' ' ' Avalable Onlne: http://saspjournals.com/sjpms 7
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 I ' I ' I ' are the prncpal moments of nerta of the traal rgd bod of mass m at ts center of mass wth a ' b ' c ' as ts aes. I' s the moment of nerta about a lne jonng the center of the rgd bod of mass m the nfntesmal bod of mass m s gven b I I l I m I n ' ' ' ' ' ' ' l ' m ' n ' are the drectonal cosnes of the lne respect to ts prncpal aes. We denote the unt vectors along the prncple aes at por p b jk the unt vectors parallel to the snodc aes b IJK wth the help of Euler's angles. The are connected b Snge Grffth (7) (959) I a b jc k J a b jc k K a b jc k a Sn Sn Cos Cos Cos a Cos Sn Cos Sn Cos a Sn Cos b Sn Cos Cos Cos Sn b Cos Cos Cos Sn Sn b Sn Sn c Sn Cos c Sn Sn c Cos. The aes O z have been defned b Szebehel [7]. Now n equaton () can be wrtten as n r n r r r ' ' A' A' A' A' A' b r r ' ' Avalable Onlne: http://saspjournals.com/sjpms 7 A A a a A A A A A b b r r A A c c A a A b A c 5R 5R 5R A A a a b A A c c (4)
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 a ' b ' c ' A ' A ' A ' (5) 5R 5R 5R R s the dstance between the prmares. The mean moton n s gven b n A A A a A A b A A c A A (6) A ' ' ' A A a A ' ' A b A ' ' A c A ' ' A. Equaton () permt an ntegral analogous to Jacob ntegral & & C. The lberaton ponts are the sngulartes of the manfold && & & f C. Therefore these ponts are the solutons of the equatons. are establshed b Sharma [4]. Let the other elements are equal to zero; We have a b c a b c the other elements are equal to zero n r n r r r r r of the bgger prmar n ths case of the smaller prmar 4 4 5 5 A A A A A A A r r 5 5 4 A' 4 A' A' A' A' A' A' r r n r n r r r r r 5 5 A 4A 4A A A A A r r 5 5 4 A' 4 A' A' A' A' A' A' r r n A A A A' A' A'. (7) TRIANGULAR LIBERATION POINTS The trangular lberaton ponts are the solutons of the equaton (7). A A' are equal to zero we smpl get r r If the values of. When ' A A aren t equal to zero we suppose that r r =. (8) Puttng the values of r r from equaton (8) n equaton () we get Avalable Onlne: http://saspjournals.com/sjpms 7
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 (9) Puttng the values of r r from equaton (8) from equaton (7) rejectng hgher order terms we get 9 7 9 A A 7 A 4 A A A 8 6 6 5 7 4 A 4 A A 6 A A () STABILITY ANALYSIS Assumng denote small dsplacement of the nfntesmal partcle from the equlbrum ponts. X X Y Y () Now o o Epng b talor s epanson consderng onl frst orders we have Where s the value of at the pont ( ) smlarl the other values the respectve values at the ponts ( ). At the equlbrum ponts we have () are Hence the equaton of moton of the nfntesmal partcle s d d n dt dt () d d n. dt dt In order to solve equaton (6) substtute t t Ae Be (4) Where AB λ are parameters. Ths gves that t t A e Bn e (5) t t A n e B e. The set of equaton (5) has nontrval soluton f Where n 4 4 (6) s defned at L 4 when the prmares are traal rgd bodes as o o o 9 9 4 4 4 4 4A 4A 7A o n 4A 4A 7A (7) Avalable Onlne: http://saspjournals.com/sjpms 7
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 o 5 5 n 7A 68A 4A 4 4 4 4 (8) 68AA 4AA 7AA o 5 7 4 4 7 A 8A 7A 4 4 6 (9) 5 8A 7A A 6 We can rewrte equaton (6) as P Q () Where P 4 n Q = The stablt of the trangular ponts requres that = must be negatve to obtan pure magnar roots.e. the dscrnnant of equaton (7) s P 4Qthat s the condton for stablt mples that: ( 7 7 ) (48 65 5 ) A (9 649 6 ) A 8 4 (86 57 5 ) A ( 5 ) A () 8 ( 7 65 788 ) A (479 98 76 ) A 6 6 If A A ( = ) are equal to zero then the stablt condton s.858965 Szebehl [7]. And f A A ( = ) are not equal to zero we suppose that crt. p A p A p A p4 A p5 A p6 A p p p p4 p5 p 6 are to be determned therefore we have 48 65 5 9 649 6 p p 7( ) 6( ) p p 86 57 5 5 p4 7( ) 8( ) 7 65 788 479 98 76 5 p6 44( ) 44( ) And the stablt condton for the trangular ponts s crt.858965. 5776A 8. 897578A 5. 5486A. 88898 A. 645894 A. 446 A. () CONCLUSION Avalable Onlne: http://saspjournals.com/sjpms 74
Elshabour SM et al.; Sch. J. Phs. Math. Stat. 5; Vol-; Issue-B (Mar-Ma); pp-69-75 We construct the locaton the stablt condton of the trangular ponts of the restrcted three bod problem wth traal prmares consderng a statonar rotatonal moton of the bgger prmar of the smaller prmar we conclude that the stablt condton s depend on that orentatons. Also we found the stablt condton n our partcular case. REFERENCES. Bhatnagar KB Gupta U; The estence stablt of the equlbrum ponts of a traal rgd bod movng around another traal rgd bod. Celestal mechancs 986; 9(): 67-8.. Gascheau M; Eamen d'une class d'equatons dfférentelles et applcaton a un cas partculer du probleme des tros corps. Comptes Rendus 84; 6:9 94. Khanna MONA Bhatnagar KB; Estence stablt of lbraton ponts n the restrcted three bod problem when the smaller prmar s a traal rgd bod the bgger one an oblate spherod. Indan Journal of Pure Appled Mathematcs 999; ():7-74. 4. Sharma RK Taqv ZA Bhatnagar KB; Estence of lbraton ponts n the restrcted three bod problem when both the prmares are traal rgd bodes. Indan journal of pure appled mathematcs ;():5-4. 5. Routh EJ; Proc. Lond. Math. Soc 875; 6:86-97. 6. Snge Grffth; Prncples of Mechancs Thrd Edton Mc Graw-Hll 959. 7. Szbehel V; Theor of Orbts Academc Press New York 967. Avalable Onlne: http://saspjournals.com/sjpms 75