PLANNING OF KEPLERIAN ORBITS: APPLICATION TO PERIODIC TRAJECTORIES

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1 ANNING OF KEERIAN ORBITS: AICATION TO ERIODIC TRAJECTORIES Vann Mical Bnntt Dpatmnt o Matmatics Univsit o Hawai i at Manoa Honolulu, HI 968 ABSTRACT Satllits av numous uss anging om t tansmission o communications to t obsvation o t at s wat. Dpnding on t unction o a givn satllit, t satllit will av a paticula obit tat suits it bst. T paticula unction o a satllit ma somtims cang, tb quiing a cang in its obit. Caul planning is a ncssit in t altation o an satllit s obit. T compl natu o tis planning quis tat it b don toug bot pul analtical tcniqus and toug appoimating numical tcniqus bo t tans o t obits. In tis pojct, implications a ound on ctain obital paamts o a piodic tajcto at making asonabl assumptions on som o t ot obital paamts tb constucting a piodic tajcto. Tis nwl constuctd piodic tajcto is tn analzd ug t to o tim-vsal smmt in dnamical sstms. T goal o tis analsis is to ind a mo gnal mtod o constucting piodic tajctois and to opull mak nw obsvations on t intconnctdnss o t vaious obital paamts. Visual aids, in t om o plots o satllit position vsus tim, povid an invaluabl pictu o actl wat t ocus o tis pojct is. Ts plots a obtaind ug t numical computing sotwa o MATAB along wit t ptinnt Kplian quations o motion and a w abita initial conditions. INTRODUCTION On Octob 4, 957, t U.S.S.R. launcd Sputnik, and bcam t ist nation to put an atiicial satllit into obit aound t at. Tis is considd b man to b t igt o t Cold a and t gun at t stating lin o t Spac Rac. T U.S.A. answd sotl tat wit Eplo, and o t nt w as t two suppows stad nck and nck, wit t U.S.S.R. a ai s widt in t lad. T Amican Alan Spad was in act t scond uman to nt spac b a m 3 das, aving bn batn b t monaut Yui Gagain, wo sd t tsold on Apil, 96. Howv, t U.S.S.R. was ovtakn on Jul, 969 wn Nil Amstong stppd om Apollo onto t suac o t moon, making a dining momnt o t t Cntu as wll as t culmination o t Spac Rac. Toda, satllits a noting mo tan a mundan alit o tcnologicall advancd nations. Ti unctions includ communications, at obsvation, wat obsvation, navigation, and connaissanc. Tis spctum o unctions quis a spctum o obits, o pats tat satllits ollow. Communications satllits must gnall maintain a id position lativ to t at in od to ctivl la signals, tis is acivd toug ollowing wat is known as a gosnconous obit. Man wat obsvation satllits collct inomation concning t nti plant twic ac da b ollowing sun snconous pola obits. Satllits a launcd wit powul ockts into spac, but it is diicult to launc tm pcisl into 3

2 4 ti dsid obits. T must to av mans o populsion to obtain ti coct pats. Omnipsnt ptubations also mak mans o populsion ncssa o pop satllit unction. T populsion mcanism must obviousl b takn into account in t planning o an obit. Tis is don wit t somtims unwild 3D contolld Kpl quations o motion. Appoimat numical solutions to tis quation obtaind ug MATAB gatl duc t tim ndd o planning, and nc simpli t nti pocss. T goal o tis pojct is to attack t poblm o planning Kplian obits om just a gl ont. T ont considd is tat o taking an initial tajcto on a givn tim intval and cating a piodic obit tat ollows tis initial tajcto o actl al o t obit. To simpli t poblm witout log signiicant gnalit, motion is considd in onl on plan. Concpts om t to o tim-vsal smmt a tn closl amind in od to dtmin ti connctdnss to tis pojct. 3D CONTROED KEER EQUATIONS OF MOTION T ollowing sstm o dintial quations togt mak up wat a known as t 3D contolld Kpl quations o motion: 3 dt d i i i & w Z C C Z Z 3 and C Z.

3 In tis quation is a unction o tim tat psnts t motion o a givn satllit in tms o t Gaussian coodinats,,,,, and. A Kplian obit is simpl an obit tat obs ts quations. Gaussian coodinats a a convnint out o dscibing llipss in t dimnsions. t is t lngt o t smi-latus ctum and nc t siz o t llips at tim t, t and t togt psnt t ccnticit and nc t sap o t llips at tim t, t and t psnt t inclination o t llips wit spct to t mutuall otogonal constant coodinat as, and t is t angula position o t satllit wit spct to t aomntiond as at tim t. and 3t psnt t populsion mcanism o t satllit wit ac unction bing t tust o t satllit in on o t mutuall otogonal dictions at tim t on diction is tangnt to t satllit s pat, on is ppndicula to t pat and in t obital plan, and t tid is ppndicula to ac o t ots. Sinc motion in onl on plan is considd, it is takn tat 3 t. Making t pvious assumption and stting qual ponding componnts o t vctos involvd in t 3D contolld Kpl quations o motion givs t ollowing somwat simpl sstm: & 3 / & [ ] & [ ] &. 3 / Solutions o tis sstm o nonlina dintial quations dscib t plana obit ollowd b a satllit wit a tust o t and t. Not tat tis simpliid plana cas o t 3D contolld Kpl quations dos not includ and anw. TIME-REVERSA SYMMETRY IN DYNAMICA SYSTEMS In t to o tim-vsal smmt in dnamical sstms, a unction R is said to b a dr vg smmt i F R, w psnts a tajcto, R psnts a dt tansomation on t tajcto, and F psnts t applicabl ocs on t tajcto. It is impotant to not tat now in tis quation is a contol unction, wic tis poblm quis. T ollowing unction R, wic acts on as wll as t contol unctions, mts t abov citia o bing a vg smmt: 5

4 R t t T. Obsv tat dr t T & & & & & & t dt T & & & & t F R t, assuming tat t ollowing conditions old: t - t - t - - t - - t - and t - -t. Tus R would b a vg smmt i it did not act on t contol unctions, o tis ason it will b calld a tim-vsal smmt o a contol sstm. SYMMETRY AND ERIODICITY Givn an abita plana tajcto dnotd b t Gaussian paamts,,, and wit contol unctions and on t tim intval om t -T to t, a mtod o inding a piodic obit tat ollows tis tajcto o al o t obit is dsid. T poblm is to ind unctions,,,,, and on t tim intval om t to t T suc tat ac o t ptinnt Gaussian paamt unctions is continuous at all points in t intval om t -T to t T and dictl om t T to t -T, and tat t simpliid plana cas o t 3D contolld Kpl quations is satisid. Obsv tat ts condition a mt b appling t unction R to t initial tajcto. T quimnt tat ac o t Gaussian paamt unctions av t sam valu at t -T and t T maintains t continuit o t satllit motion. Tis continuit condition quis tat T - -T and T -T. Futmo, it is takn tat, o as o andling t quations. Satllits cannot jump toug spac, and tis insus tat an solutions tat do jump toug spac a liminatd. An unction suc tat - is said to b vn t gaps o vn unctions a smmtic about t ais o t dpndnt vaiabl, in tis cas t ais. Assuming tat allows o t computation o in tms o, b t tid quation divd om t simpliid plana cas o t 3D contolld Kpl quations. It can b sown in a staigtowad mann tat tan. Tis gatl simpliis t numical intgation o t sstm, wic is pomd in t ollowing sction. AICATION AND DEMONSTRATION Givn an abita al obit on an intval [-T,], t mtods dscibd abov dscib ow to tun to t initial coniguation on t tim intval [,T], and to pat t 6

5 ccl indinitl. An abita st o data tat could appl to man modn da satllits was n to dmonstat visuall t implications o t unction R abov. T 3D contolld Kpl quations w intgatd ug MATAB and t abita st o data was usd o t initial conditions o t vaious dintial quations. Figu is t initial al obit stating at tim t -T and nding at tim t. T objct locatd insid o t tajctois psnts t at. On unit on ac igu psnts mgamts, o ^7 mts. T unction R was tn applid to tis al obit in od to obtain Figu, stating at tim t and nding at tim t T. Not tat t nd o t pat in Figu is continuous wit t bginning o t pat in Figu, and tat indd t obit is piodic. Figu : t initial al pat. Figu : t calculatd tun tajcto. CONCUSION aps t most impotant pat o tis pojct is not t discov o t tim-vsal smmt R o tis contol sstm, but t intptation o R. Figu abov is a conct ampl o an abita al pat. Howv it could b an continuous pat and t application o R to t Gaussian paamt unctions would ild t smmtic pat tat compltd t obit into a piodic on. Tis tcniqu could pov to b quit usul in t planning o Kplian obits. T act tat nal an pat could b ollowd and subsquntl compltd into a piodic obit as man implications. It is impossibl to imagin t numb o asons w a satllit migt b quid to ollow a paticula pat. But wit tis vsal-smmt R o tis contol sstm, a mtodical wa o dciding wat to do at t pat is ollowd as bn dtmind. T implications o t tim-vsal smmt R o tis contol sstm owv a quit limitd. T simpliication o t t-dimnsional poblm into a two-dimnsional poblm is signiicant and sould not b ovlookd. V al would a satllit qui planning in a gl plan. T limination o t tid contol unction 3 gatl simpliid t sstm o nonlina dintial quations, paticulal b liminating and om t ptinnt sstm. I 3 w again considd, t diicult in solving t 3D contolld Kpl quations would gow ponntiall. Not onl would ac o t dintial quations psnt 7

6 now b complicatd wit t addition o and, but two ntil nw dintial quations would appa, tos bing t two associatd wit t divativs o and. ACKNOEDGEMENTS T auto would lik to tank is mnto D. Moniqu Cba along wit D. Tomas Habkon, wo was vital o t numical intgation wit MATAB. T auto would also lik to tank t Hawai i Spac Gant Consotium o giving im tis oppotunit to pinc t sac pocss. REFERENCES Hai, E. 4 Solving Odina Dintial Equations, Sping Vlag, Blin. amb, J., Robts, J. 997 Tim-vsal smmt in dnamical sstms: a suv. sica D. ollad, H. 966 Matmatical Intoduction to Clstial Mcanics, ntic-hall, Inc., Nw Js. Sontag, E. 998 Matmatical Contol To: Dtministic Finit-Dimnsional Sstms, Sping Vlag, Sdn. 8