Invariant Relative Orbits Taking into Account Third-Body Perturbation
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1 Appled Matemats - ttp://dx.do.og/.6/am..8 Publsed Ole Febuay (ttp:// Ivaat Relatve Obts Takg to Aout Td-Body Petubato Wald Al Raoma Glles Mets Depatmet of Astoomy Faulty of See Cao Uvesty Cao Egypt Obsevatoe de la Côte d Azu Gasse Fae Emal: wald_aoma@yaoo.om Reeved Otobe ; evsed Deembe 9 ; aepted Deembe 7 ABSTRACT Fo a satellte a obt of moe ta 6 km alttude te effets of Su ad Moo o te obt a t be eglgble. Wokg wt mea obtal elemets te seula dft of te logtude of te asedg ode ad te sum of te agumet of pegee ad mea aomaly ae set equal betwee two egbog obts to egate te sepaato ove tme due to te potetal of te Eat ad te td body effet. Te expessos fo te seod ode odtos tat guaatee tat te dft ates of two egbog obts ae equal o te aveage ae deved. To ts ed te Hamltoa was developed. Te expessos fo te o-vasg tme ate of age of aoal elemets ae obtaed. Keywods: Ivaat Relatve Obts; Td-Body Petubato; Hamltoa. Itoduto Fomato flyg s a key teology eablg a umbe of mssos w a sgle satellte aot aompls: fom emote sesg to astoomy. Te elatve moto w sows o dft eve pesee of a lage dstubae ould be a vey attatve soluto. To mata te fomato ad ostellato te elatve dfts due to te petubato betwee te spaeaft sould be aefully osdeed. Ivaat Relatve Obts sows o dft betwee te spaeaft due to te petubato eve f pesee of a lage dstubae. Te lteatue s wealt wt woks dealg wt desgg eta vaat elatve obts fo spaeaft flyg fomatos ad t seems wot to sket some of te most elevat woks. Saub ad Alfed [] peseted a metod to establs vaat elatve obts fo spaeaft fomato flyg applatos. Tey desged elatve obt geomety usg dffeees mea obt elemets. Two ostats o te tee mometa elemet dffeees ae deved. Zag ad Da [] emoved te dfts by adjustg te sem-axs of te followe satellte ad obtaed a smla oluso. By meas of Rout tasfomato ad dyamal system teoy Koo ad Masde [] developed a metod to fd te vaat obt. Te ad [] ad Meg et al. [5] oluded fom te pot of vew of elatve obtal elemets tat te dfts of elatve obt esult fom te obtal lato ad gt aseso of asedg ode of te two satelltes. Bggs ad Beea [6] poposed a me- tod to detemate te vaat obt wt te leade s obt of zeo lato based o te tagetg metod aos dyams. Abd El-Salam et al. [7] used te Hamltoa famewok to ostut a aalytal metod to desg vaat elatve ostellato obts due to te zoal amos ; ; up to te seod ode assumg beg of ode. Ou popose was to exted Saub ad Alfed [] ad Abd El-Salam et al. [7] model by addg te effet of te td body w ave mpotat at g alttude. Usg te Hamltoa famewok te petubatos a be easly added. Te Hamltoa of te poblem was ostuted by osdeg te effet of te td body of. Te expessos fo te tme ate of age of te seula elemets ae obtaed seod ode odtos ae establsed betwee te dffeees mometa elemets (sem-majo axs eetty ad lato agle) tat guaatee tat te dft ates of two egbog obts ae equal o te aveage.. Hamltoa Appoa Tee ae seveal ways to deve te equatos of moto fo ay su system. We empaszed o te Hamltoa stutue fo ts system. Te Hamltoa fomulato allows fo addtoal osevatve foes to be added to te Hamltoa tus te addto of omplexty to te model a be opoated wt ease. No-osevatve foes a be added te mometa equatos of moto. Te Hamltoa equatos of moto allows us to detly Copygt SRes.
2 W. A. RAHOMA ET A. use otol ad smulato teques. Notatos te wole text we use te well-kow keplea elemets: te sem-majo axs a te eetty e te lato te gt aseso of asedg ode te agumet of pegee ad te mea aomaly M. We also use te tue aomaly f ad a - temeday vaable e. Te Hamltoa te peset famewok a be wtte te fom p U U t () wee U s te foe futo due to te Eat s gavtatoal potetal ad p s te aoal mometum veto ad Ut te dstubg futo due to te effet of petubg body... Ifluee of Oblateess Petubatos Te atual sape of te Eat s tat of a eggplat. Te ete of mass does ot le o te sp axs ad ete te meda o te lattudal otous ae les. Te et esult of ts egula sape s to podue a vaato te gavtatoal aeleato to tat pedted usg a pot mass dstbuto. Te Eat s gavtatoal potetal s usually expessed by te followg expesso (Vt s potetal) µ Re m U P Cm m Sm m s os s m wee R e s te equatoal adus of te Eat µ gm e s te Eat s gavtatoal paamete wee g s te gavtatoal ostat; ae te geoet oodates of te satellte wt measued east of Geew; C ad ae amo oeffets; m S m m P s ae assoated egede Polyomals. I te potetal futo te tems wt m m ad m oespod espetvely to zoal tesseal ad setoal amos. Te Eat gavtatoal potetal a be ewtte up to seod ode tuatg te sees at as Abd El-Salam et al. [7] µ U S S f Re os Re 5S Ss f 5S s f 8 Re S 5S S S 5S os f 6 () wee S s ad s te zoal amo oeffets... Td Body Petubato Te effet of te td body te moto of a atfal satellte ave beame patulaly teestg ow we spae debs mposes a seous teat to spae atvtes. Tese petubatos ae te most mpotat measm of delveg majo Eat obtg objets to te egos wee te atmospee a stat te deay. If t s assumed tat te ma body; Eat; wt mass me s fxed te ete of te efeee system x-y. Te petubg body wt mass m s a ellpt obt wt sem-majo axs a eetty e ad mea mo- to gve by te expesso a gm m e ad ae te adus vetos of te satellte ad m (assumg ) ad s te agle betwee tese adus vetos. Te dstubg futo (usg te tadto expaso egede polyomals) due to te td body s gve by Domgos et al. [8] wee wt Ut m m m e a a os a ad osos f s f () f f osos os ss f f sos os oss Copygt SRes.
3 W. A. RAHOMA ET A. 5 Usg te Delauay aoal-vaables defed by l M Mea aomaly a lggh R R A R A A K 5 6 e e e Te Hamltoa Equato () up to te seod ode g Agumet of te Pegee G e a ow be expessed as a powe sees as follows ogtude of asedg ode H G os Cosdeg as a small paamete of te poblem te odes of magtude up to te seod ode of te () volved paametes ae defed as follows: ; ; ad let us defe te dmesoless paametes as lem s te wee epesets te upetubed pat of te pob- petubato: 5 A a os 6 S S f A a 8 5S Ss f 5S s f 8 A a S 5S S S os f 5S os f 6 a Ka os f os f s f s f a Now we eed to elmate te sot as well as te log peod tems of te satellte moto addto to te sot peod tems of te dstae petubg body. Usg te petubato teque based o e sees ad e tasfom Kamel [9] te tasfomed Hamltoa fo dffeet odes a be wtte as Abd El- Salam et al. [7] ad Domgos et al. [8]. ** ** ** (5) wt ** A ** G A H H H H H H 6µ G G G G G G G ** G G 5 7 G k A H H H H G H (5 ) 8 G G G G G G H G wee K 5 k e 8 µ e 8 Usg te Hamltoa aoal equatos of te moto to wte agumet of mea lattude ( ) s te sum of te mea aomaly ad te agumet of pegee (.e. l g) as ** ** ** ** l g G G wt G ** ** (6) Copygt SRes.
4 6 W. A. RAHOMA ET A. A H A G G ** os 7 A 5H A G G G ** 5os 7 ** A H H H H H H G G 6µ G G G G G G G A H H H H G H 8 G G G G G G G k 6 A 5 os 5os os 6 os 5 5 8os 5os 7 6µ A 8 k 5 5 os 75os os 5os 6 os 9 7 ** 6A H H H H H H G G G 8µ G G G G G G G 5A H H H H G 6k G 5H 8 G G G G G G 8 5 6A 5 7 8os 77 os 8os 5os 5 6os 5os 8 8µ 5A k 8 7os 85os 9 6os 89os 6 5os 5 ad te seula dft ates of te logtude of te asedg ode : wt ** ** H H ** H ** H os A A 5 7 H G (7) ** A H H H H 5 7 G 9 5 G 5 9 H 6µ G G G G 5A H 5 G G 5 G 5 ( G 7H ) 6k H A os os 5 7 os os 9 5os 8 6µ 5A os os ( 7 os os ) 6kosos. Costats fo Ivaat Obts I ode to pevet two egbog obts fom dftg apat te aveage seula gowt eeds to be equal. Sot peod osllatos a be goed ee se tese ae oly tempoay devatos. Te log peod ates ap- pea seula ove a few weeks ad tey ae. Copygt SRes.
5 W. A. RAHOMA ET A. 7 Se te mea agle quattes lg ad do ot detly otbute to te seula gowt te values a be ose at wll. Howeve te mea mometa values Gad H (ad teefoe mpltly ae ad ) must be aefully ose to mat te seula dft ates. To keep te satelltes fom dftg apat ove tme t would be desable to mat all tee ates lg. We mpose te odto tat te elatve aveage dft ate of te agle betwee te adus vetos be zeo. Ts esults j (9) Now ad j j j () a be ewtte as 5 mt t mt t osm () mt 5 mt t mt t os m () mt mt mt mt wee te o-vasg oeffets mt ad ae omputed Appedx I. et te efeee mea obt elemets b e deoted wt te subspt. Te dft ate of a egbog obt a be wtte as a sees expaso about te efeee obt elemet ee t s eoug to keep te seod ode oly as xx xx xx! xx xx xx xx xx x x xx xx xx! xx xx xx xx xx x x () () wee we make use of te fat tat ad oly also supposg t at s te dffeee mea lattude ates x ad x Note tat ts t eoy wll lead to a aalytal seod ode odtos o te mea obt elemets. To establs a moe pese set of obt elemets satsfyg Equatos (9) ad () ete o ould be ose ad te emag two mometa obt elemet dffeees foud toug a umeal oot solvg teque. Howeve te aalytal seod ode odtos povde easoably auate solutos to tese two ostats equatos ad povde a wealt of sgt to te beavo of Eat potetal ad td body effet vaat elatve obts. Te equed devatves a be evaluated as 5 mt mt t mt t osm 5 mt mt mt t mt t t t osm 5 m m mt t mt t t s 5 6 os 5 mt mt t mt t m Copygt SRes.
6 8 W. A. RAHOMA ET A. ad 5 mt mt t mt t t t tt osm 5 mt m m mt t mt t t osos 5 mt mt t mt t t t osm 5 m mt mt t mt t t t sm 5 mt mt m m mt t mt t s 5 mt mt t mt t osm 5 mt mt t t mt t t osm m 5 mt mt t mt t t sm 5 m mt mt t mt t 6 os 5 m mt t mt t t t tt os m mt 5 mt mt t mt t t osm 5 mt t mt t t t osm m 5 mt mt mt t mt t t t sm 5 m mt t mt t sm wee x ad xx wt x. x x x To efoe equal dft ates ad betwee egbog obts we must set ad paded Equatos () ad () yelds equal to zeo ex- (5) (6) Equatos (5) ad (6) ae two smultaeous ol- equatos tee ukows amely ea algeba. We oe of tese tee ukows s assumed kow (say ) tese two equatos a be solved as: Multplyg Equato (5) by ad Equato Copygt SRes.
7 W. A. RAHOMA ET A. 9 (6) by ad te subtatg yelds wee 5 5. ( 7) Substtutg Equato (7) to Equato (5) yelds a algeba equato of fout degee oly te fom wee 5 d d d d d d d (8) d 5 5 d d Soluto of te Quat Equato (8) Te oots of te quat Equato (8) a be wtte as wee wt wee ad ab abd d d a b efg efg d efg+ d e a d d * d l f g j 8b d d d d d j m m 5 d dd d d d d l m d 9d d d 7d d 7d d 7d d d. 5 5 Substtutg te fou oots s to Equato (7) yelds te fou ostats s tat guaatee te vaae of te elatve moto of eta satellte ostellato a b ab b a a b ab a b d abbdad a b d abd Coluso Auate modelg of elatve moto dyams fo tal odtos lose to te leade satellte s essetal fo flyg fomato. Teefoe te solutos of teest ae estted to a spef set of tal odtos tat lead to peod moto su tat te satelltes do ot dft apat. Ts pape sowed a aalytal expesso to seula dft ates due to oblate Eat model tuatg ts pote- tal sees at ad td body effet ad set t equal betwee two egbog obts. It followed te same steps used befoe Abd El-Salam et al. [7] fo te Eat model so te alulato of Abd El-Salam et al. [7] ad Saub ad Alfed [] s a speal ase fom ts al- lato ( ) a be ulatos. Te vaato te ose at wll fo te omal lato ad te vaatos bot te eetty ( e ) ad sem-majo axs Copygt SRes.
8 W. A. RAHOMA ET A. ( a ) fom te omal values ae set to zeo. Noted tat tese ostat odtos ae ot justfed ea te tal lato agle. Usg stead of e to avod te sgulaty we e but fo e te osgula elemets must be used. Futue developmets of ts appoa to te fomato flyg poblem lude aote petubato foes lke sola adato ad lusola effets. 6. Akowledgemets Te fst auto ws to expess s appeato fo te suppot povded by te Fe govemet ude te No de dosse: 6888B No afflato: 96/777. Te autos gatefully tak efeees fo te elpful suggestos ad ommets. REFERENCES [] H. Saub ad K. Alfed Ivaat Relatve Obts fo Spaeaft Fomatos Celestal Meas ad Dyamal Astoomy Vol. 79 No. pp do:./a:687 [] Y. Zag ad. Da Satellte Fomato Flyg wt Petubato oual of Natoal Uvesty of Defese Teology Vol. No. pp. 6-. [] W. S. Koo ad. E. Masde Dyams ad Fomato Flgt Poeedgs of AIAA Gudae Navgato ad Cotol Cofeee Moteal August p. 9. [] X. ad. Study o Relatve Obtal Cofguato Satellte Fomato Flyg Ata Meaa Sa Vol. No. 5 pp do:.7/s9--9- [5] X. Meg. ad Y. Gao Petubato Aalyss of Relatve Obts Satellte Fomato Flyg Ata Meaa Sa Vol. 8 No. 6 pp [6]. D. Bggs ad V. M. Beea A Sea fo Ivaat Relatve Satellte Moto t Woksop o Satellte Costellatos ad Fomato Flyg Sao ose dos Campos 5 pp. -. [7] F. A. Abd El-Salam I. A. El-Toamy M. K. Amed W. A. Raoma ad M. A. Rassem Ivaat Relatve Obts fo Satellte Costellatos: A Seod Ode Teoy Appled Matemats ad Computato Vol. 8 No. 6 pp. 6-. do:.6/j.am.6.. [8] R. C. Domgos R. V. demoaes ad A. F. Pado Td-Body Petubato te Case of Ellpt Obts fo te Dstubg Body Matematal Poblems Egeeg Vol. 8 8 p.. do:.55/8/7665 [9] A. A. Kamel Expaso Fomulae Caoal Tasfomatos Depedg o a Small Paamete Celestal Meas ad Dyamal Astoomy Vol. No. 969 pp do:.7/bf888 7 Appedx I 9 A A A A A A A A 5A 99A 5A 95A 575A A 5 99A 95A 567A 85A A 5A 95A 5A 75A 575A k k 6 k k 9k A 75A 675A 5A 55A 5A 7A A 5 A 5 A 5A k 8k Copygt SRes.
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