HREE DIMENSIONAL GEOMERY MAINENANCE FOR FORMAION FLYING ON ELLIPIC ORBIS akanao SAIKI ), Koch NASUME ) and Jun chro KAWAGUCHI ) ABSRAC ) Mtsubsh Havy Industrs, Ltd. ) Mtsubsh Elctrc Co. ) Japan Arospac Exploraton Agncy (JAXA), Hgash anaka, Komak 458-856, Japan E-mal: takanao_sak@mh.co.jp hr has bn mpndng ntrst n th formaton flyng wth many satllts. Multpl satllt systm nhancs th mssons' flxblty wth lss total mass and cost, and ralzs som mssons that wr mpossbl wth a sngl satllt. At th Insttut of Spac and Astronautcal Scnc (ISAS/JAXA), th plasma and magntc fld obsrvaton mssons wth svral satllts s undr nvstgaton. h msson undr consdraton s dsgnatd as SCOPE. h obsrvaton ara of th SCOPE msson s twnty or thrty arth rad away from th cntr of th arth whr th gomagntc fld has ntracton wth th nrgtc partcls from th sun. hrfor ts orbt bcoms hghly llptc. hs papr frst dscusss th dsgn mthod for spontanous mantanng th formaton gomtry on th llptc orbts. In th obsrvaton aspct, th formaton of plural satllts s rqustd to consttut a polygon that assurs th hgh spatal rsoluton obsrvaton. hs study nxt show th orbtal dsgn mthod for th SCOPE msson. h frozn proprty that mantans hgh spatal rsoluton nar th apog s found fasbl for llptc orbt. Numrcal xampls ar prsntd wth practcal llustratons. NOMENCLAURE a : sm-major axs : ccntrcty : nclnaton Ω : rght ascnson of ascndng nod ω : argumnt of prg t : prg passag tm t : tm f : tru anomaly M : man anomaly n : man moton p : sm-latus rctum µ : gravty constant r : rlatv poston
. INRODUCION In rcnt yars, thr has bn mpndng ntrst n th formaton flyng wth many satllts. Multpl satllt systm nhancs th mssons flxblty wth lss total mass and cost, and ralzs som mssons that wr mpossbl wth a sngl satllt, for xampl, nfrard ray or lasr ntrfromtrs, mult pont magntc fld obsrvaton, larg spac antnna and so on. At th Insttut of Spac and Astronautcal Scnc (ISAS, JAXA), th plasma and magntc fld obsrvaton msson wth multpl satllts s undr consdraton. h msson undr consdraton s dsgnatd as SCOPE (SCOPE: Scal COouplng n Plasma Envronmnt), th succssor msson to GEOAIL. GEOAIL has obsrvd th plasma surroundng of th arth, but t cannot dstngush th tm fluctuaton wth spcal dstrbuton bcaus t s a sngl satllt msson. o ovrcom ths dffculty, n th SCOPE msson, th obsrvaton of many ponts usng th plural satllts s rqurd. In th obsrvaton aspct, th formaton of plural satllts s rqustd to consttut a polygon that assurs th hgh spatal rsoluton obsrvaton. h obsrvaton ara of th SCOPE msson s twnty or thrty arth rad away from th cntr of th arth whr th gomagntc fld has ntracton wth th nrgtc partcls from th sun. hrfor ts orbt bcoms hghly llptc. On such hghly llptc orbt, t s vry dffcult to kp th gomtry of formaton flyng bcaus th rlatv postons of th satllts chang largly as tm gos on. On possbl soluton to mantan th satllts gomtry s postv control of rlatv poston wth satllts ful. But t s unralstc bcaus th ful consumptons of satllts probably bcom larg. So th dsgn of th orbt that spontanously kps th hgh spatal rsoluton s ncssary. It s known that th thr-dmnsonal formaton mantnanc ovr th orbt s mpossbl. But, as th focusd ara of SCOPE msson s nar th apog, th dsgnng th orbt that kp th polygon bcoms possbl. hs papr frst dscusss th dsgn mthod for spontanous mantanng th formaton gomtry on th llptc orbts. h rlatv poston of satllts on th llptc orbt can b xprssd wth th small dffrncs of th Kplran paramtrs. By gvng th adquat valus to ths paramtrs, w can dsgn many ntrstng orbts. h orbt whch can mantans th dstanc btwn th satllts or two-dmnsonal gomtry can b dsgnd. And ths study nxt show th orbtal dsgn mthod for th SCOPE msson. h frozn proprty that mantans hgh spatal rsoluton nar th apog s found fasbl for llptc orbt. Numrcal xampls ar prsntd wth practcal llustratons.. RELAIVE POSIION ON ELLIPIC ORBIS Frst, th coordnat systm s dfnd as Fgur. X I Y R Z R r X R prg Z I f r rf rfrnc orbt cntr of th arth Fgur : Rfrnc orbt and rotatng coordnat
h orbtal paramtrs of th rfrnc satllt on th rfrnc orbt ar ( a,,, Ω=, ω =, t ). h rotatng fram s consdrd hr; X R axs s th radus vctor and Z R axs ponts th angular vlocty drcton. h poston vctor of rfrnc satllt s p r rf = ( r,,), r=. () + cos f Hr th satllt whos orbtal paramtrs ar ( a,,, Ω, ω, t ) s consdrd. Whn ths satllt s locatd nar th rfrnc satllt, ths orbtal paramtrs should b wrttn as follows; whr a = a+ δ a, = + δ, = δ, t = t + δt, () Ω + ω = π k+ δω ( k =, ±, ±,...) δ a <<, δ <<, δ <<, δω <<, δt <<. () h poston vctor of ths satllt n th rotatng fram can b obtand. f ' s tru anomaly and whr r ' r sat = RZ ( f ) RZ ( Ω ') RX ( δ ) RZ ( ω ' + f '). (4) r ' s radus of satllt rspctvly. rrf rrf rrf r = rrf + δ a+ δ + nδ t, (5) a n t f f f f f f f ' = f + δ a+ δ + nδ t. (6) a n t f f f rrf rrf n( t t) sn f rrf rrf µ =, = acos f, = sn f a a t p, (7) f n( t t ) a f (+ cos f )sn f =, =, a rrf. (8) f µ = ( cos f ) + t p hn th rlatv poston of satllt, r= r r. (9) sat rf can b drvd as follows; µ r n( t t ) sn f p a rf x= aδ cos f + δtsn f + δ a. ()
δ µ y= r rf (+ cos f )sn f + r rfδω+ rrf δ t (+ cos f ) p. () n( t t ) a δ a r rf z= r δsn( f Ω '). () rf. ONE/WO-DIMENSIONAL GEOMERY MAINANANCE In th prvous scton, th rlatv can b rprsntd by orbtal paramtrs. By usng ths rsults w can dsgn th orbts that can mantan th on/two dmnsonal gomtry.. Innr Product of Rlatv Poston Wth th varabl convrson of δ µ α =, β = δ t, γ = δω, δ = δ cos Ω ', ε = δ sn Ω', () p th nnr product of rlatv poston can drvd as followng form; whr r r j j j = + k cos + k r rf k= p j p p kf q sn kf, (4) 4 5 7 αα ββ γ γ j = + + + + + + ( βγ + β jγ ) + δδ + εε j 5 7 p = αα + + 4 ββ + γ γ j βγ + β jγ ( ) 4 5 αα ββ βγ β jγ ( ) j p = + + + δδ + εε j p = αα + ββ j q = + ( αβ j + α jβ) + ( αγ + α jγ ) j q = ( αβ j + α jβ) + ( αγ + α jγ ) ( δε + δ jε ) j q = ( αβ j + α jβ). (5) h numbr of trms s svn (Whn th rfrnc orbt s crcl, th numbr of trms s fv). 4
. Dstanc Mantnanc In gnral, t s dffcult to mantan th dstanc btwn th satllts on llptc orbts. But by usng q. (4), w can dsgn such orbts. Whn th dstanc btwn th satllts s rqustd to b kpt at D, th followng quaton should b satsfd; D p + p cos kf + q sn kf =. (6) k k k= rrf By substtutng q. () to q. (6), th rght-hand trm can b transformd. D D D D = cos cos + + f + f. (7) rrf p p p hn, th condtons for dstanc mantnanc bcom as follows; D D D p = +, p =, p =, p = q = q = q =. (8) p p p As th numbr of orbtal paramtr s 5 and th numbr of condton s 7, th soluton dos not xst. By nglctng th coffcnts of th hgh frquncy trms, p and q, th approxmat soluton can b obtand. D α =, β =±, p 4 7 + 8± 4 9 + 4. (9) 4 4 9 γ = + β, δ = + β, ε = 4 4 4 hs soluton xsts only undr th followng condton; < (9 + ).766. () 6 Fgur shows th rsults of numrcal smulaton. Whn th ccntrcty s small, th dstanc s almost constant. In ths study, th orbt that mantans th dstanc btwn th satllts s drvd analytcally, but anothr approach s dscrbd n rf. [] and [4]. 5 4 =. =.569 rang 99 98 97 96 95 6 8 4 6 tru anomaly [dg] Fgur : Smulaton rsults of dstanc mantnanc 5
. Smlar Gomtry Mantnanc In ordr to mantan th two-dmnsonal smlar gomtry, th followng condtons ar consdrd; r r r r =, r = r. () h formr condton mans th mantnanc of th angl of rlatv postons and th lattr mans th qualzaton of th dstanc. h followng condtons can b drvd from q. (); p = p = = q =, p = p, p = p,, q = q. () As th numbr of orbtal paramtr s and th numbr of condton s 4, th soluton dos not xst. By nglctng th hgh frquncy trm and ntroducng two fr paramtrs, th solutons ar obtand. α= σ cos φ, β = σ sn φ, γ = + σ sn φ, 4 δ=± σ sn φ, ε= cosφ 4 4. () π π π α = σ cos φ+, β σ sn φ, γ σ sn φ, = + = + + 4 π π δ =± σ sn φ, ε cos φ 4 + = + 4 By usng ths paramtrs, th two-dmnsonal gomtry can b mantand. h rsult of numrcal smulaton s shown n Fgur. Although th sz changs, th lttr E form s mantand. f= [dg] f=45 [dg] X f=9 [dg] 5 5 5 Y -5-5 -5 - - f= X [dg] - - f=5 [dg] - - f=8 [dg] - 5 5 5-5 -5-5 - - - - - - Fgur : Mantnanc of E formaton. 6
4. HREE DIMENSIONAL GEOMERY MAINANANCE FOR SCOPE In ths scton, th thr dmnsonal gomtry mantnanc mthods ar rprsntd. 4. Rlaton btwn th Spatal Rsoluton and Gomtry h objctv of SCOPE msson s to nvstgat th spatal structur of plasma and magntc fld. In short, th stmaton of spatal drvatvs of varous obsrvatons s rqurd. At last four satllts ar ncssary for stmaton of frst ordr spatal drvatvs. And th accuracy of stmaton dpnds on th gomtry. For xampl, whn th four satllts ar placd on th sam plan, th stmaton of spatal drvatvs s mpossbl. hrfor, th gomtry s vry mportant for spatal obsrvaton. Hr w assum that th valu of obsrvaton s th functon of satllt s poston. Whn th spatal drvatv Y = f ( x, y, z). (4) f f f f =,, x y z s unform on th obsrvaton ara, th obsrvaton valu of satllt on r = ( x, y, z) can b wrttn as follows; Y = Y + f (5) r. (6) Y s th obsrvaton valu at th orgn. Hr w consdr on mothr satllt on th orgn and thr daughtr satllts. h dffrnc of obsrvatons btwn mothr and -th daughtr satllt s Z = Y Y = r f. (7) hn, th f can b stmatd as follows; Z = = ˆ f = r r r. (8) Whn th accuracy of obsrvaton dpnds on th dstanc btwn th satllts, wrttn as follows; whr s Z can b Z = Y Y = r f + r v, (9) v s wht nos whos varanc s P= r r = r σ. In ths cas, th covaranc of stmaton valu σ. () h gomtry that mnmz th J = r( P) s th good formaton for th spatal obsrvaton and t s shown n th fgur 4 (lft-hand fgur). If th obsrvaton dffrncs btwn daughtr satllts ar avalabl, th bst gomtry for spatal obsrvaton s ttrahdron shown n fgur 4(rght-hand fgur). 7
Fgur 4: Sutabl formaton for spatal obsrvaton 4. Rght-Angld Baslns Hr w consdr th formaton that consttuts thr baslns whch cross prpndcularly (Fgur 4, lft). W can asly consttut a basln n th drcton of Z axs by a satllt whos nclnaton s dffrnt for a whl from th mothr satllt. So what w should do s to consttut two baslns whch cross prpndcularly on th orbtal plan of th mothr satllt. h rlatv poston of th daughtr satllt on th orbtal plan of th mothr satllt s dpndnt on thr paramtrs, δ, δω and δ t. Hr nw paramtrs R, φ and k ar ntroducd. δ µ µ = Rsn φ, δ t = Rcos φ, δt δω= kr. () p p hn th rlatv poston can b wrttn as follows; r cosφ cos( f π φ) cos( f π / ) = R (+ cos f ) + + k. () r sn φ sn( f π φ ) sn( f π / ) As ths quaton ndcats, th paramtr R dtrmns th sz of rlatv moton and both φ and k dtrmn th form of rlatv moton. As dscrbd abov, t s mpossbl to mantan thr-dmnsonal formaton ovr th orbt. Howvr, t s not ncssary to mantan th formaton at all th placs on orbt n plasma physcs as th focusd ara s nar th apog. hn hr w dsgn th orbt that mantans th thr-dmnsonal gomtry at 7 f 9 [dg]. h orbtal paramtrs of th rfrnc orbt (mothr sat.) ar shown n abl. abl : Orbtal paramtrs of mothr satllt radus of prg r p : R radus of apog r a : R sm-major axs a : 57 ccntrcty :.88 nclnaton : [dg] R : radus of arth For th dsgn of th orbt that mantans two rght-angld baslns, sx paramtrs should b consdrd ( R, R, φ, φ, k and k ). o rduc th numbr of paramtrs, th followng condtons ar consdrd; 8
R = R = R, φ = φ = φ, k = k = k. () hs quatons ar gvn from th symmtry of two rlatv poston vctors at th apog. hrfor w can dsgn th formaton shap by consdrng only two paramtrs, φ and k bcaus R dtrmns only th sz of formaton. By calculatng φ and k that mnmz 9 π J = θ ( θ : angl btwn two baslns). (4) f= 7 w can obtand dsrd orbt. φ =8.5 [dg] and k =.94 ar obtand undr th condtons of abl. h rsults of numrcal smulaton ar shown n Fgur 5 and Fgur 6. Fgur 5 shows th hstors of angl and th rang of th baslns. h angl s kpt at about 9 [dg]. Fgur 6 shows th rlatv postons of th daughtr satllts at f = 7, 8 and 9[dg] and th rlatv moton n nrta systm. h rght trangl can b mantand n nrta spac. 4 rang(sat.) rang(sat.) angl rang 9 angl [dg] 8 7 75 8 85 9 7 tru anomaly [dg] Fgur 5: Rang and angl of th two baslns Y 5 5 f=7[dg] daughtr sat. daughtr sat. morhr sat. X 5-5 f=8[dg] d m d -5 5 5 m d -5 - -5 d f=9[dg] Y - f=7~9[dg] daughtr sat. morhr sat. daughtr sat. - X Fgur 6: Mantnanc of rctangular trangl 9
4. trahdron Formaton In ths scton, th ttrahdron formaton mantnanc s ndcatd. As dscrbd n prvous scton, thr satllts on th sam orbtal plan and on out-of-plan satllt ar consdrd. For th mantnanc of ttrahdron formaton, thr satllts should consttut rgular trangl and out-of-plan satllt should b locatd abov th cntr of th trangl. Hr w assum that th orbtal paramtrs of out-of-plan satllt ar a ' = a, ' = + δ, ' = δ, Ω ' =± 9[dg], ω ' = 9[dg], δ t ' =. (5) By consdrng q. () and (5), th shap of formaton can b dtrmnd by th four paramtrs, φ, k, δ and δ. As dscrbd n scton 4., th ndx of spcal rsoluton s P= r r r = r. (6) So, by calculatng th paramtrs that mnmz 9 J = P, (7) f= 7 orbt that mantand th ttrahdron formaton s obtand. Fgur 7 shows th rsult of numrcal smulaton. Although thr s som dstorton, th rgular ttrahdron gomtry s kpt nar th apog. Z - Y - f=7[dg] - X Z - Y - f=75[dg] - X Z - - Y f=8[dg] - X Z - - Y f=9[dg] - X Fgur 7: Mantnanc of ttrahdron formaton
5. SUMMARY hs papr shows th mthod of th orbtal dsgn for SCOPE msson. SCOPE msson rqurs th hghly llptc orbt for th obsrvaton. As s wll known, t s mpossbl to mantan th thr dmnsonal formaton ovr th orbt, but th frozn proprty that mantans hgh spatal rsoluton nar th apog s found fasbl for llptc orbt. h orbts dsgnd n ths papr ar vry ffctv for many scnc mssons as wll as SCOPE msson. REFERENCES []. Sak and J. Kawaguch, Orbtal Dsgn and Control Stratgy of th Formaton Flyng n Ellptc Orbt, IAC--A..5,. [] K. Natsum,. Sak and J. Kawaguch, On th Formaton Flyng wth Frozn Gomtry n Ellptc Orbts wth Applcaton to Gomagntc Plasma Physcs Msson, IAC--A..7,. [] Z. an and P. Banum, An Improvd Stratgy for Mantanng Constant Dstanc btwn Satllts n an Ellptcally Orbtng Constllaton, AAS/AIAA Spac Flght Mchancs Mtng, [4] P. Banum, A. Strong and Z. an, chnqus for Dployng Ellptcally Orbtng Constllaton n Along-rack Formaton, IAC--A..5,. [5] S. Nakasuka, Y. Kawakatsu,. Nnomya,. Fujwara and. Nakamura. Study onth Rlatv Moton and Orbt Dsgn for Clustrd Satllts, Procdngs of 4th Workshop on Astrodynamcs and Flght Mchancs, 994, pp. 88-9.