The Study of the Asymmetric Multiple Encounters Problem and Its Application to Obtain Jupiter Gravity Assisted Maneuvers

7/30/2019 The Study of the Asymmetric Multiple Encounters Problem and Its Application to Obtain Jupiter Gravity Assisted Ma

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Hndw Pshn CopotonMthmtc Poms n EnnnVom 2013, Atc ID 745637, 12 pshttp://dx.do.o/10.1155/2013/745637

Research ArticleThe Study of the Asymmetric Multiple Encounters Problem andIts Application to Obtain Jupiter Gravity Assisted Maneuvers

Denilson Paulo Souza dos Santos, Antnio Fernando Bertachini de Almeida Prado,

and Evandro Marconi Rocco

Division o Space Mechanics and Control, INPE, C.P. 515, 12227-310 Sao Jose dos Campos, SP, Brazil

Cospondnc shod ddssd to Antono Fnndo Btchn d Amd Pdo; [email protected].

Rcvd 7 Novm 2012; Rvsd 31 Mch 2013; Accptd 3 Ap 2013

Acdmc Edto: Sv M Gtt Wnt

Copyht 2013 Dnson Po Soz dos Sntos t . Ts s n opn ccss tc dsttd nd th Ctv CommonsAttton Lcns, whch pmts nstctd s, dstton, nd podcton n ny mdm, povdd th on wok spopy ctd.

T Mtp Enconts Pom s dscd n th tt s th pom o fndn tjctos o spcc tht vsom moth pnt, dscs tjctoy n th ntpnty spc, ndthn os ck to th moth pnt. T psnt ppxtnds th tt ndth dptnd v ns o th spcc nzd to nonsymmtc. T sotons shown n tms o th t (]) nd ccntc nomy (). T vocty vton () qd o th tns s so shown. Tn,ths stdy s nzd to consd th possty tht th spcc mks cos ppoch wth th moth pnt to chnts ny n th tn tp. T vocty (

) nd ny vton (

) d to ths pss otnd. T topcs stdd h

cn ppd n mssons tht v nd com ck to th Eth, wth th o o stdyn th ntpnty spc, s w s omssons whos ojctv s to mk n tton n th ny o th spc vhc thoh swn-y wth th moth ody.

1. Introduction

T tt s xtnsv wth spct to poms nvovntns ots nd optm spcc mnvs. Goddd[1], n 1919, ws pon n stdyn th pom otnsn spcc twn two ponts. In hs sch,ppoxmtd sotons poposd o th pom osndn ockt to hh ttds mnmzn th consmd. Rdn ot tnss, cssc wok ws

md y Hohmnn [2], n 1925. T pom o tnsn spcc twn two copn cc ots wth tm ond cnt oc fd (Nwtonn) ws stdd,otnn s soton -mpsv ptc tns ot.Ts soton s sd vn tody, o fst msson nyss,nd ws consdd s th fn soton o th pom p toth y 1959, whn Shtnd [3] nd Hok nd S [4]showd tht th-mpsv tns s mo conomc,dpndn on th fn nd nt ots nvovd. Svoth pps stdyn mpsv mnvs to chn thot o spcc cn ond n [519]. Lt, th do nonmpsv thst ws consdd to sov ths otmnv pom. Lwdn [20, 21] dscd thotc

compt soton o th pom o ot tns twntwo ponts wth mnmm consmpton. Ts pomcvd th nmLwdns Pom d to ts commtmntto sov t. H ntodcd nw concpt, th pm vcto,whch s th Ln mtp ssoctd wth th vocty

vcto, to fnd ncssy condton o optm mpsvtjctos ccodn to th mntd o ths vcto. Atht, mny sts ppd n th tt td to thstyp o mnvs, s shown n [2234].

Rdn th Mtp Enconts Pom, t ws nto-dcd n th tt y Hnon [35], n 1968, tht stdd mnv wh th spcc stts ts moton n moth ody nd thn tns to ths sm ody,

jony n dnt tjctoy wth spct to ths mothody. H dmonsttd tht th soton cn dcd tosmp c qtons. Ts mnv cm knowns Hnons ns (Pdo nd Bock [36, 37]). Otso ths typ w so xmnd y Bno [38], Pko[39], nd Htz nd Hnon [40, 41] consdn th ccstctd th-ody pom s th mthmtc mod. Inptc, [40, 41] nvsttd th mpotnc o th o oth Jcon constnt n ms o ots o th cs

= 0


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2 Mthmtc Poms n Ennn

nd th stty n th vcnty o th ots wth = 0, wh s th mss o th scondy ody n cnonc nts. Tysoht to fnd consctv mtns twn 3 nd thot o2 n th stctd th-ody pom o th cs = 0. Lt, How [42] nzd ths stdy to ncdptc ots o th pms, nd Sntos [43] nd Sntos

t.[44] consdd ns wth dnt mntds o thnt nd fn ponts o th tns.T pom stdd h s th tns o spcc

om on ody ck to th sm ody, wth possswn-y n th tn pss o th spcc. T nmos ppctons o ths pom, sch s () to tns spcc om th Eth to n ntpnty tp nd thnck to th Eth, wthot th nd o mnvs dn thspocss, mpyn th optmzton o th consmpton;() to tns spccom th Moon,whch my ncd pss y th Eth, nd thn pt t ck on th Moon;() ndzvos mnvs, whn on dss tht th spc

vhc stnds onsd noth spcc; (v) to mnvth spcc so tht t vs nd tn to th Eth topom swn-y to chn ts ny, nd thn o to thot So Systm; (v) to mov spcc tht s ond pnt, k Jpt, to noth octon n th So Systm ovn ot to th ntst spc.

Fo ths st two ppctons, t s ncssy to comnth Mtp Enconts Pom wth th vty ssstdmnvs. Ts typ o mnv s sd vy on nntpnty tjctos. It ss cos ppoch wth cst ody to chn th ot o spcc. Rncs[4561] show som ppctons o tht pom. Dts oth comnton o thos two poms tt xpndt n ths pp.

2. Description of the MultipleEncounters Problem

o dsc th Mtp Enconts Pom, t s c 1nd 2 th two pmy ods o th systm, wth msss(1 ) nd , spctvy. 2 s n cc ot (nth on vson o th pom stdd y Hnon [35])o n n ptc ot (n th xtnson md y How[42]) ond 1. Hnon [35] stdd ths pom ndpshd sotons o th cs o cc ots. How [42]pshd sotons o th ptc cs, wh th tnssw symmtc wth spct to th ppss. Pdo nd

Bock [36, 37] so pshd sotons o ths pom, nth sm sttons, sn th Lmt mthod to sov thpom.

T spcc 3 (th thd ody) vs 2 t th pont ( = 0), oows tjctoy ond 1, nd thn mtsn wth2, t th pont ( = ), wh 0, [0, 2]wth > 0 (F 1). T tt sy stdd thspom nd th condton = 0, t now ths stdys xtndd to consd th css wh 0 nd notncssy symmtc [43]. A po, t s ssmd tht2 =0 (dsd th cts o th ttcton o2 on 3), whchdcs th pom o th ods to th two-ody pom.So, t s poss to s Kps qtons n th dvopmnt

3

2

1

2

1

2

r1 r2

Figure 1: T omty o Hnons pom omtd s Lmtpom [43].

o th sotons. wo mpss w sd to pom thtns mnv; th fst on mks th spcc to vth moth ody, nd th scond on s sd to cpt thspcc n. F 1 stts ths stton, wh th-- systm o nc s ny nt systm. T3 odydos not scp om th vtton ttcton o1 ndhs n mss.

Lmts pom cn omtd s oows. Fndn nptd ot nd th mthmtc mod vny w tht woks wth th nvs sq o th dstnc

(Nwtonn omton), tht conncts two vn ponts nd , wth th tns tm () spcfd. In th tt,sv schs hv sovd ths pom sn dstnctomtons. Fom omtc nyss o th pom[36, 37], t s poss to wt th qtons shown nxt.

() T poston coodnts o2 dfnd s (ssm-n cc ots, n th ocntc systm)

2 = cos ,2 = sn . (1)

() T poston coodnts o3 dfnd s (ssm-n copn ptc ots)3 = cos ,3 = 1 2 sn , = 3/2 sn ,

(2)

wh (, ,nd) thKpn mnts smmjoxs,ccntcty nd ccntc nomy o th ot o th ody,nd

s th tm.


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Mthmtc Poms n Ennn 3

T dnt possts o th ot o3 ond1 [44]. Anyzn thm, t s poss to dfn [36, 37] = { +1,1, th ppss s n n scss { postv,ntv,

= { +1,1, th ot sns s {possv

,tod, = { +1,1, th pss y = 0 s t th { ppss,popss.

(3)

Usn ths omton, th nzd poston coodntso3

3 = ( cos ) ,3 = 1 2 sn ,

= 3/2

(

sn ) .

(4)

o sov th pom t s ncssy to mpos tht (1) nd(4) hv th sm vs, cs 2 nd 3 n th smntposton.Rmm tht th n vocty ()othsystm s nt; so, cn consdd to th tm s ws th n. So, = nd = (ccntc nomy), wh

cos = ( cos ) ,sn = 1 2 sn ,

= 3/2 ( sn ).(5)

Ts st o qtons w sd to nt th so-tons o th pom. Hnons pom [35], omtd sLmts pom, cn dscd n th wy shown nxt[36, 37].

() T poston o3 s known t = 0 (pont , ntpont o th tns ot). T poston 1 cn spcfds ncton o th n 0 y

1 = (1 2)

1 + cos 0 . (6)It s th sm v o 2 nd 3, cs 2 nd 3occpy th sm poston t th nt momnt (

= 0).

() T poston o3 s so known t = (pont, th fn pont o th tns ot). T poston 2 sdscd y th qton

2 = (1 2)

1 + cos () . (7)() T tot tns tm s vn y = 0.(v) T tot n , twn th ponts nd ,

o th cs wh th ots ptc, hs sv possvs.

Fst o , t s ncssy to consd two poss chocso th tns; th fst on ss th sns o th shotst

poss n twn nd (shot wy), ndth scondon ss th sns o th onst poss n twn thstwo ponts (on wy).

A consdn ths two chocs, t s so ncssyto consd th possts o mtvoton tnss. Inths cs, th spccvs th pont, mks on o mocompt votons ond1, nd thnt osto th pont. Ts, th possts, comnn thos two ctos,ndct tht th vs o [ 0 + 2] nd[2 0 + 2]. T s no pp mt o , ndths pom hsn nfnt nm o sotons, xcptn thcs wh th ot o3 s poc o hypoc, wh hs nq v.

Lmts pom soton s th Kpn ot thtcontns th ponts nd nd tht qs th vntns tm = = 0 o th spcc to tvtwn thos two ponts. In ths pp, Goodns Lmtotn s sd to sov Lmts pom [62].

Poss ppctons o ths tchnq ntpntysch n th So Systm, ss o tnspottonsystm twn th Eth (1) nd th Moon (2) whno ot cocton s qd, nd so oth.

o t th vocty vton (), th oown stps sd.

(1) Fnd th d nd th tnsvs componnts oth vocty vcto o 2 t th ponts nd . Ty so th vocty componnts o3 mmdty oth fst mps nd mmdty th scond mps,spctvy, d to th ct tht th otsntcpt on thosponts. T oown stps sd to t th [ 43]

() d vocty

= sn

]

1 2 ; (8)() tnsvs vocty

= 1 + cos ] 1 2 , (9)wh ] s th t nomy.

(2) Sov Lmts Pom o tns twn thponts nd. Atths nstnt th tns tm s so known.

(3) A otnn th componnts o th vocty vctommdty o nd th mpss, t s poss to

cct th mntd o th two mpss (1 nd 2)nd to dd thm to otn th tot mps qd otns ().3. Numerical Solutions for the Multiple

Encounters Problem

o mk nmc smtons, th pocd s to vy thnt nd fn postons o 2 o th mnv. Tn,th vs o th vocty vton nd oth qntts cn cctd to fnd th mnvs wth th mnmmconsmpton. T mk th sts s to vsz, t sntstn to show th sts consdn fxd v o


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0. Tn, th sotons vn n tms o ( [0, ]), whch s th n tht dfns th fn pont o thmnv. s th mxmm v o. T n o thspmts s povdd s n npt o th smton. Fo chst o th smtd mnvs, th fs show th sotonso th Mtp Enconts Pom.

3.1. Simulation or the Parameters = 2.5 and = 3.5 rad. T pocd o th smton s thoown.

(1) Vs o th ccntcts o th pms, 1 nd2, dfnd. It mns tht th dynmc systms dfnd, cs th mnvs ssmd to pnnd th smmjo xs s wysnty, y thdfnton o th cnonc nts.

(2) An nt v o th pont wh th spccvs th ody2 s sctd. In th xmp shownh, ths v s = 2.5 d. Smtons wthdnt vs w md, nd th sts w vysm.

(3) T n tht dtmns th pont o th tno th spcc to 2, , s thn vd to covth dsd n o ths v. Ts condtonso dfns th dton o th tns, cs thn vocty s nty; so, 0 so psntsth tm o th tns.

(4) Usn th tm o th tns nd th nt ndfn ponts dfnd , t s poss to sov thssoctd Lmts pom. Ts soton vs thtns ot nd th tot ncmnt o vocty topom th tns.

(5) Consdn th possty o sv votons oth tns, my o sotonspps och po ponts. Ty pottd n fs tht show thcompt st o sotons.

Lt s ssm th oown pmts: = 2.5; 3.5 d. F 2 shows th sotons n tms o tht nomy (]) oth stton wh th ccntctyo thpmy (2) s 0.5. T hozontxsshows thv ond,ndth vtc xsshowsth t nomy oth fnpont o th mnv. Evy dot n ths pot psnts onsoton o th pom nd so on tns ot. Not thtth sotons not nq, cs th spcc my v

sv votons ond th pmy. Up to 6 votonsw ncdd n ths smtons.

T sm sotons cn sn n F 3 t now ntms o th ccntc nomy o th sm sttonwh th ccntcts o th pms 0.5. T ho-zont xs n shows th v o n d, nd th vtcxs now shows th ccntc nomy o th fn pont o thmnv. Evy dot n ths pot so psnts on tnsot nd so on soton o th pom.

F 4 shows th tot vocty vtons o th smsotons, wh th ccntcts o th pms 0.5.T hozont xs shows n th v o n d, ndth vtc xs now shows th tot mps tht nds to

2

1

0

1

23

2 1 0 1 2 3

4

5

6

7

8

Trueanoma

ly()(ra

d)

(rad)

Figure 2: Soton n tms o th t nomy (]); = 0.5, 2.5 3.5 d.

1

0

1

2

3

4

5

6

7

2 1 0 1 2 3

Eccentricanoma

ly()(ra

d)

(rad)

Figure 3: Soton n tms o th ccntc nomy (); = 0.5,2.5 3.5 d.

ppd to th spcc to compt th mnv. Evy dotn ths pot so psnts on tns ot. It s ntstnto not th n o vs o th tot vocty vton(

), whch os om vs cos to zo to vs s hh

s 7 cnonc nts. It s mpotnt to not tht th vs ovy n to zo; so, th tns om on odyck to th sm ody n thos condtons hs vy smmps to pomd.

F 5 shows th sm-mjo xs o th sm so-tons. T hozont xs shows n th v o nd, nd th vtc xs now shows th sm-mjo xs oth tns ot. Evy dot n ths pot so psnts ontns ot.

Sv tns tms ( = ) w smtd,snc th nt n ws kpt constnt. Amon nmostns ots, t s shown tht th ms o sotonswh th

qd o th tns s vy sm, cos


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0

1

2

3

4

5

6

7

2 1 0 1 2 3

(canonicalunits)

(rad)

Figure 4: Vocty vton () vss , o = 0.5, =

2.5d, nd

2.5

3.5d.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Sem

imajoraxis()(canonica

lunit)

2 1 0 1 2 3

(rad)

Figure 5: Smmjo xs () vss , o = 0.5, = 2.5 d,nd 2.5 3.5 d.

to zo. Ts ots ood sotons, cs th consmpton o ths tns s mnm. Sotons wthmnmm

(cos to zo) povd tjctos tht

ood cnddts o t mssons.Sm smtons w pomd o dnt vs

o th ccntcty. T sts sm, nd so thy omttd h.

T sts o thos smtons wth vos ccntc-ts showd tht systm o pms wth sm ccn-tcts tt o th xcton o Mtp EncontsOts, wth spct to th consmpton, cs thy sm whn compd to systm o pms wth hhccntcts ( 1). Anoth pont s tht th moccntc th ot o3, th hh th consmpton oth spcc to xt nd tn to th ody2. Ts mpstht t s mo common to fnd tt sotons (mnm

)

able 1: Sotons wth < 0.1.Eccntcty (d) nomy Eccntcnomy Voctyvton

] (d) (d)

= 0

0.001 1.9990 1.9990 0.000111

2.001 0.9990 0.9990 0.0001670.933 1.9330 1.9330 0.0001790.099 0.9010 0.9010 0.000195

= 0.10.001 1.9988 1.9986 0.0001762.001 0.9988 0.9986 0.0002090.002 1.9975 1.9973 0.0003532.002 0.9975 0.9973 0.000419

= 0.52.001 0.9965 0.9940 0.0089400.001 1.9965 1.9940 0.0089401.999 1.9965 1.9980 0.0089461.999 0.9965 0.9980 0.008964

= 0.9 0.000 2.0000 2.0000 0.0079842.000 1.0000 1.0000 0.0079842.000 1.0000 1.0000 0.0122482.000 2.0000 2.0000 0.012248

mon cc ots ( = 0) o mon ots wth smccntcts.

1 shows sotons wh th vocty vton ()o th Mtp Enconts Pom s mnm consdnccntcts om zo to 0.9. T xstnc o sotons sc wth sm vton o vocty n sttons, t thmnmm vs occ o ow vs o th ccntcty oth pms.

4. The Swing-By Maneuvers

T dynmcs o th two ods s sd n th psntomton. T systm s consdd to omd y thods. It s poss to sy tht

(1) th ody1, wth fnt mss, s octd n th cnto mss o th Ctsn systm;

(2) 2, sm ody, cn pnt o stt o1,n Kpn ot ond 1;

(3) ody 3, spcc wth nfntsm mss, stvn n Kpn ot ond 1, whn tmks n ncont wth 2.

Ts ncont chns th ot o3 wth spct to1 nd, y th hypothss ssmd o th pom, t sconsdd tht th ots o1 nd 2 do not chn.

Usn th ptchd concs ppoxmton, th qtonstht qnty thos chns v n th tt [50].T stndd mnv cn dntfd y th oownth pmts (F 6):

() ||, th mntd o th vocty o th spccwth spct to 2 whn ppochn th cstody;


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3

12

+

Figure 6: Swn-y mnv.

() , th dstnc twn th spcc nd th cs-t ody dn th cosst ppoch;

() , th sot v o th vocty t th ppss;(v) , th n th ppoch, tht s, th n twn

th n connctn th pms nd th ppss oth cos ppoch tjctoy.

Hvn thos vs, t s poss to otn , tht s,h o th tot dcton n, y sn th qton [50]

= csn 11 + 2/2 . (10)A compt dscpton o ths mnv nd th dv-

ton o th qtons cn ond n Bock [50]. T fnqtons podcd s oows:

= 22 sn sn , (11) = 22 sn sn , (12)

= = 2 sn = 2 sn . (13)4.1. Hypotheses or a Swing-By with Jupiter (JGAJupiterGravity Assisted) Obtained by Solving the Multiple EncountersProblem. o sov ths pom, th oown ssmptons md.

(1) T systm s omd y two ods n ptc otsond th mt cnt o mss nd thd mss-ss ody movn nd th cton o th vttonocs.

(2) T on o th systm s pcd n th cnt omss. T hozont xs s th n connctn thods 1 nd 2 nd th vtc xs s ppndc- to tht n.

(3) T spcc vs th pont (s F 1),cosss th hozont n (Sn-Jpt), os to thpss, nd thn chs th pont , wh th cosppoch occs (s nowF 6).

(4) A th cos ppoch, th spcc modfs thvocty, ny, nd n momntm (11) to (13).

(5) W sd th cnonc systm o nts. Ts om-ton mps tht th nt o dstnc s th dstnctwn 1 (Sn) nd 2 (Jpt); th n

vocty () o th moton o1 nd 2 s ssmdto nty; th mss o th sm pmy (2) svn y = 2/1 +2 (wh 1 nd 2 th msss o1 nd 2, sp.), nd th mss o2s (1), to mk th totmsso th systm nty;th vtton constnt s on.

In ths systm o nts, th vtton pmt o

Jpt s

= 9.47368421 102. Mo dts v

n [43].

4.2. Swing-By with = 0 .1, = 2.5, and 3.5 rad. Nmc smtons w pomd snJpt o th ody om wh th spcc scps ndpoms th swn-y n th tn tp. Ts stton cnhppn n pctc stton wh t s dsd to s thvty o Jpt to hp to mov spcc tht s ondJpt to noth octon n th spc, k noth pnt,stod, nd so oth. T vocty () nd th ny ()

vtons, whch dpnd on th fn n o th mnv, nyzd th Jpt Gvty Assstd (JGA).Amon th smton pmts, th ppoch dstnc oth spccdn th swn-y s n mpotnt pmt.In od to t th stonst poss sts wthot tknth sk o codn th spcc nto th tmosph oJpt, v o 1.2 Jpt ds 5.725019385545 104 ws sd. 2 shows th sotons wth mxmm

vtons o vocty nd ny.T nmc sts show tht th st n o th

vocty vton () dos not mpy th st n nny (), whch s consqnc o (11) to (13). Not thtth vton n ny hs n xt tm (sn), ctnth mntd o th chn n ny nd not th mntdo th chn n vocty. Ts n s dtmnd y thomty o th ppoch, whch s dfnd y th chocs

o th nt nd fn ponts o th Mtp EncontsMnv. F 7 shows th vton o vocty ()otnd om th swn-y o th stton wh thccntcty o th pms s 0.1, th nt poston o thmnv s spcfd y th n = 2.5 d,nd thfnposton o th mnv (n ) s n th n twn2.5 nd 3.5 d. Not tht th postons wth vy shpmxmms. F 8 shows th qvnt vton o thny (). Not tht, n n, th mxmms n thsm poston, t th sht dncs twn othpots, o th sons dy xpnd. Vyn th vs oth ns (0), vs o mxmm ns o th vton oth ny (

) nd vocty (

) w ond. T vton


7/13


able 2: Swn-y wth > 1.2, , nd n cnonc nts. = 0.1

= 2.5 d An o ppoch Vton o vocty Vton o ny = 3.5d d D d D2.999 171.830043 3.8745 221.992498 1.2159 0.81346

3.001 171.944634 3.8770 222.135737 1.2144 0.814721.001 57.3530753 3.8902 222.892041 1.2053 0.820421.999 114.534263 3.8795 222.278977 1.2119 0.815322.001 114.648855 3.8801 222.313354 1.2116 0.815622.999 171.830043 3.8607 221.201816 1.2222 0.805053.001 171.944634 3.8607 221.201816 1.2222 0.805091.999 114.534263 3.8882 222.77745 1.2113 0.822662.001 114.648855 3.8871 222.714425 1.2119 0.822092.999 171.830043 3.8857 222.63421 1.2125 0.821211.999 114.534263 3.8900 222.880582 1.2067 0.821132.999 171.830043 3.8857 222.63421 1.2125 0.821213.001 171.944634 3.8817 222.405027 1.2146 0.819131.999 114.534263 3.8900 222.880582 1.2067 0.821131.999 114.534263 3.8882 222.77745 1.2113 0.822662.001 114.648855 3.8871 222.714425 1.2119 0.822092.999 171.830043 3.8607 221.201816 1.2222 0.805053.001 171.944634 3.8607 221.201816 1.2222 0.80509

0.2

0

0.4

0.6

0.8

1

1.2

2 1 0 1 2 3

SB

(canonica

lunits)

(rad)

Figure 7: Vton o vocty () otnd y th swn-y oth stton wh = 0.1, = 2.5 d, nd 2.5 3.5 d.

o th vocty () cn so vszd n 2. It svs tht th sv ponts o mxmm ns.

Fs 9 nd 10 show th nc twn th voctyvtons. T tot s th dnc twn th vnd om th swn-y nd th v spnt to mnvth spcc. F 9 shows tht th som ntvsotons. It mns, physcy, tht th vton o vocty to

0.5

0

0.5

1

SB

(canonica

lunits)

2 1 0 1 2 3

(rad)

Figure8: Vton o vocty () otnd yth swn-y n thstton wh = 0.1, = 2.5 d, nd 2.5 3.5 d.

mnv th spcc s thn th n otnd omth swn-y nd th mnv shod not pomd.Postv vs mn tht th mnv shod don. Ocos, t s ntstn to ook o th sttons wth mx-mm v o ths mnv. F 10 s n mpfcton oF 9, consdn ony th postv sts.


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0.05

0.1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

total(canonica

lunits)

234 1 0 1 2 3 4

(rad)

Figure 9: ot (SB ) otnd y th swn-y o thcs wh = 0, = d, nd d.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

23 1 0 1 2 3

total(canonica

lunits)

(rad)

Figure 10: Postv nc n , otnd om th swn-y nth stton wh = 0, = d, nd d.

In th smtons pomd n ths stdy, som stsv n th tt [50], whch hps n vdtn thcctons pomd h w so confmd.

() I th swn-y occs hnd th ody2 (180


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able 3: Swn-y wth > 1.148. = 0.9

= 2.5 d An o ppoch Vton o vocty Vton o ny = 3.5 d d D d D2.292 131.32193 4.82564 276.48892 1.15561 1.14820

2.293 131.37922 4.82347 276.36436 1.15533 1.148212.294 131.43652 4.82130 276.23991 1.15505 1.148212.295 131.49381 4.81913 276.11552 1.15478 1.148212.296 131.55111 4.81696 275.99125 1.15451 1.148212.297 131.60841 4.81479 275.86709 1.15425 1.14820

0.4

0.2

0

0.2

0.4

0.6

0.8

0 50 100 150 200 250 300 350

SBswing-by

(canonica

lunits)

Te angle of the approach () swing-by

Figure 12: otnd y th swn-y o th stton wh =0.1, = 2.5 d, nd = 3.5 d.

n ppndcto th tjctoy o2. Ts ct ntsvs n/2 o 3/2 o , whch nts vso th vton o ny. o vy ths xpctd sttonoccs, stdy s md ssmn 0.9 o th ccntctyo th pms. Fs 13 nd 14 show th sts. Fo

v o th ccntcty o th pms o 0.9 nd fxdpont to stt th mnv ( = 2.5 d), th vtono ny (F 11) s shown s ncton o th pont tofnsh th mnv (

). It s notcd tht th mxmm

vs chd s thn n th stton wh = 0.1(F 8).F 14 s scy th vton o ny so s

ncton o th pont to stt th mnv (), n th-dmnson ph. It s notd tht th f s scycomposd o sm pn fs tht nd ch oth. Ts s th mn son why th psnt ppconcntts n shown sts o fxd v o th pontto stt th mnv (). It s mch s to vsz thsts, nd th s no oss o nomton. 3 showsth dtd sts o th ponts wth mxmm vtono ny. It s ntstn to not tht th ncs o thccntcty o th pms ncss th cost o th nt

0.8

0.6

0.4

0.2

0

0.2

0.40.6

0.8

1

1.2

2 1 0 1 2 3

SB

(canonica

lunits)

(rad)

Figure 13: Vton o ny () otnd y th swn-yn thstton wh = 0.9, = 2.5 d, nd 2.5 3.5 d.

mnv o th spcc, t t so ncss th nsom th swn-y. T nxt st shows tht th ns nd th ncs o th ccntcty s nfc o thwho mnv.

5. Conclusion

Optmm spc mnvs w stdd, whch th o ofndn tjctos tht dcs th consmpton ontpnty mssons. In ptc, n sttons wh

th spcc vs nd coms ck to th sm ody(Mtp Enconts Pom), consdn nonsymmt-c sttons o th stt nd fnsh o th mnv. Svsmtons w pomd nd shown, nd th sotonsond ow-cost mnvs tht cn sd o th pnnno mssons.

Tn, th stdy ncdd swn-y mnvs n thtn pss y th moth ody s om o nnny. T nmc smtons showd tht th vocty() nd th ny () vtons ncs wth th ccn-tcty o th ot o th pms. Ts ct s ntstn,cs t contsts wth th ct tht th ncs o thsccntcty nts nt mnvs wth hh costs.


10/13


SB

(canonica

lunits)

0.5

0

0.5

1

2

2

0

0 12 32

1 (rad)

0 (rad)

Figure 14: Vton o ny () otnd y th swn-y n thstton wh

= 0.9,

2.5

3.5d, nd

2.5

3.5 d.

T smtons showd ons wh th nt st spostv.

It ws so shown tht th vty ssstd mnvsmd wth th Jpt Pnt (JGA) cn povd consd- vton o vocty nd ny o spcc, thsdcn th costs o th msson. In th fs nd tsshown n ths wok, t s vfd tht ths mnv s pow too tht cn so sd n ntpnty mssonstht q tht th spcc vs ody nd tns tto ths sm ody. Ts s ntstn o mnv tht

nds to mov spcc tht s ond Jpt to snd tto noth octon o th So Systm o yond.

Acknowledgments

Ts wok ws ccompshd wth th sppot o SoPo Stt Scnc Fondton (FAPESP) nd Contcts2009/16517-7, 2011/09310-7, nd 2011/08171-3, CAPES, CNPq(contct 304700/2009-6), nd INPENton Insttt oSpc RschBz.

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