Earth Escape Trajectories Starting from L2 Point

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1 Earth Escap Trajctoris Starting from L Point Masaki Nakamia (Soknai Univrsit), Hiroshi Yamakawa (AXA) Abstract Th L an L points of th Sun-Earth sstm, which ar locat about.5 million km awa from th Earth in th sun an anti-sun irction, ar wil us as suitabl locations for spac obsrvatoris to a grat xtnt. Ths points ar also thought as a gatwa for th intrplantar transfr in th nar futur. This papr invstigats th scap trajctoris from th Earth s gravit laving th Sun-Earth L point in th rstrict thr-bo problm. Firstl, w iscuss th onimpuls scap stratg assuming vlocit corrction onl at L point, an th two-impuls scap stratg assuming th first vlocit corrction at L point an th scon vlocit corrction at th subsqunt prig point. B kping th hprbolic xcss vlocit constant, th rquir vlocit corrction amount an th flight path gomtr ar analz both analticall using acobi intgral. Finall, optimal scap trajctoris of ths stratgis ar calculat b SQP (Squntial Quaratic Programming mtho), optimal multi-impuls scap stratg is also invstigat.. INTRODUCTION Thsas, u to incrasing intrst in scintific xploration of plants an astrois, optimal trajctoris to ths clstial bois hav bn invstigat xtnsivl. In gnral, ths xplorations ar in n of a larg amount of ful for spaccraft, an various stuis pursuing low nrg transfr trajctoris hav bn prsnt. -6 Although th arth scap starting from th low arth orbit was thoroughl invstigat in Rf. 7, th scap from th Earth s sphr influnc starting from L an L point is not t full unrstoo. Th L an L points of th Sun-Earth sstm ar goo positions for spac obsrvatoris bcaus an objct aroun L an L points can maintain th sam orintation with rspct to Sun an Earth. Svral astronom satllits hav thn alra utiliz ths points i.., Halo/Lissajous orbits. In aition, transfr to th intrior an xtrior rgion of th arth is rlativl as b aition of a littl nrg at L an L points. Thrfor, ths points ar also consir as a caniat of gatwa for th intrplantar transfr in th nar futur (Rfs. 8 an 9). For ths rasons, th purpos of this papr is to invstigat th arth scap trajctoris laving th Sun- Earth L point, which ma wll b consir as th initial partur squnc of th intrplantar transfr. Th planar circular rstrict Sun-Earth-spaccraft (S/C) thrbo problm is assum for th Sun-Earth gravit mol. Th rquir vlocit corrction amount, th tim of flight an th flight path gomtr ar invstigat. Sinc w can trat th namics starting from Sun-Earth L point similarl to that of L, w concntrat on th orbital namics starting from L point in this papr. Matsumoto t al. 0 xamin th low-thrust arth scap from Sun-Earth L point using Elctric Dlta-V Earth Gravit Assist (EDVEGA), bas on th rsult of impulsiv scap manuvrs b fixing th rquir vlocit corrction amount. Whil, this papr fixs th arth scap nrg (C3) an minimizs th vlocit corrction amount. It is bcaus th arth scap nrg (C3) an th scap irction from th Earth ar mainl trmin b th intrplantar path sign. This stu iscusss firstl two arth scap stratgis. On is th on-impuls scap that is th irct scap with V onl at L, an th othr is th two-impuls scap assuming V at L point an V at th subsqunt prig point. Ths two stratgis ar analz using th acobi intgral to stimat th minimum rquir vlocit corrction, an thn ar optimiz numricall to vrif th analtical stimation of th rquir vlocit corrction an to xmplif th gomtr of th scap trajctoris. Finall, th multiimpuls scap stratg is iscuss.. DYNAMIC MODEL Th scap from th Earth s gravit starting from th L point is invstigat assuming th planar circular rstrict Sun-Earth-S/C thr-bo mol (Fig. ). Lt m an m b th masss of Sun an Earth rspctivl. Th origin of th rotating fram sstm is locat at thir mutual cntr of mass, an x-axis is th lin from Sun to Earth. Th mass of m an m, th angular vlocit an th istanc from Sun to Earth ar normaliz to on, an th coorinats of th Sun an th Earth ar (-µ, 0) an (- µ, 0), whr µ m ( m + m ). Th quations of motion for S/C ar givn b U U x + x, + x (),() x whr U is a potntial function as follow. µ µ U (3) S E Th following quation fins th acobi intgral, which is a consrvativ quantit trmin from th initial conitions, µ µ ( x + ) + x S E (4) Fig. Sun-Earth lin fix rotating fram.

2 In th abov-mntion mol, thr ar fiv points whr solar gravit, Earth gravit an cntrifugal forc acting on S/C ar balanc. Th ar call libration points. Th L point of th Sun-Earth sstm is locat about.5 million km awa from th Earth in th anti-sun irction. In this stu, this L point is st as th partur point of S/C. 3. ASSUMPTIONS In this papr, th raius of Earth s gravitational sphr is assum km bon th L point, an th hprbolic xcss vlocit v is fin at this istanc. Nxt, th maximum flight tim from libration point L to th bounar of th Earth s gravitational sphr is fin as on ar hr to avoi xtrml long flight tim. Aitionall, w assum that th rquir scap C3 is.0 km /s fin at th bounar of th Earth s gravitational sphr in consiration of transfr to th nar arth astrois with chmical an/or th low thrust propulsion sstm an transfr to plants with low thrust propulsion sstm. Th cass with C3 othr than.0 km /s wr also stui but omitt hr for brvit. Th C3 is th masur of th nrg rquir for th chosn intrplantar trajctor an is fin as th squar of th hprbolic xcss vlocit as follows. C 3 v i. µ E V (5) whr V i. is th vlocit of S/C on th bounar of Earth s gravitational sphr in gocntric inrtial sstm, µ E is th gravitational constant of Earth an E. is th istanc from Earth to S/C (i..,.0 x 0 6 km).thrfor, V. of.8 km/s is rquir to satisf th conition C3 i V.0 km /s. Th i. is xprss b position an vlocit in rotating fram as, ξ cost sin t x ( µ )sin t V i. η sin t cost + x ( µ )cost (6) Fig. shows th scap vlocit vctors, V, in rotating r. fram as a function of th scap position an irction in th Sun-Earth lin fix rotating fram. As can b sn from Eq. 6, th magnitu of scap vlocit in th rotating sstm with constant C3 valu of.0 km /s pns on th scap vlocit irction φ at a crtain scap. Fig. shows that th magnitu of V taks th smallst r. valu whn th scap vlocit irction is normal an countr-clockwis to th raial irction an th biggst valu is obtain whn th scap vlocit irction is normal to an clockwis with rspct to th raial irction. Th rang of V magnitu is approximatl from 0.8 to r..6 km/s. Fig. Escap vlocit vctors in rotating fram as a function of th scap position an irction in th Sun- Earth lin fix rotating fram (C3.0 km /s ). point at an angl of θ L b prforming th vlocit incrmnt V at L. An th S/C scaps from th Earth s gravit irctl. Th scap vlocit is fin as v at th Earth s gravitational sphr with an anglθ in th rotating sstm, whr x-axis is fin as zro gr. an b th acobi intgral valus at L with Lt L V S/C initial stat (i.., zro vlocit) an aftr hav bn appli, rspctivl. Not that th S/C position os not var uring th impulsiv manuvr; Eq. (4) ils An th scap vlocit Intgral (Eq. 4) as follows, V L (7) v is xprss with th acobi µ µ v x ( x ) S. (8) whr with simpl gomtrical consirations (cosin formula) it is foun that x + µ ) + ( µ ) cos( π θ ) (9) ( S. + cos( π θ ) (0) At this point, C3.0 km /s is obtain at th Earth s gravitational sphr b incrasing th S/C vlocit v with th aition of V at L. Th vlocit corrction V in th on-impuls scap stratg is simpl V. () xprss as 4. ESCAPE APPROACH acobi Intgral Analsis for On-Impuls Escap In this sction, th on-impuls scap stratg is analz with acobi Intgral. Fig. 3 illustrats th onimpuls scap stratg mol. Th S/C starts form L Fig. 3 On-impuls scap mol in th rotating sstm.

3 Optimal Escap Trajctoris of On-Impuls Escap Stratg Nxt, th scap trajctor of on-impuls scap stratg is optimiz b SQP (Squntial Quaratic Programming mtho) in consiration of th following factors. min Objctiv function: Paramtr: magnitu of first vlocit corrction V, partur anglθ L Constraint conition: C3.0 km /s, scap Th rsults of th optimal scap trajctoris b SQP ar shown in Figs. 4 an 5. Fig. 4 shows th xampls of optimal scap trajctor. Fig. 6 shows th optimal minimum rquir V b SQP an th analticall riv V b th acobi Intgral from Eqs. (5) through () at ach scap position. It is foun that th optimal minimum rquir V is smallst whn th scap position is aroun 0 gr. Also, th minimum rquir V is changing rapil aroun 90 gr bcaus th scap vlocit irction (th scap trajctor) is chang whthr S/C coms clos to th Earth onc or not (i.. irctl scap). An thn, Fig. 6 shows th rlation btwn rquir V an th scap vlocit irction. It is foun that it can scap from Earth s gravitational sphr to all irction. Fig. 5 Comparison of th analticall riv with optimal rquir V (on-impuls scap). V Fig. 6 Rlation btwn optimal rquir V an scap vlocit irction (on-impuls scap stratg). Fig. 4 Exampl of th optimal on-impuls scap trajctoris in th rotating fram. acobi Intgral Analsis for Two-Impuls Escap In this sction, th two-impuls scap stratg is invstigat with acobi Intgral. Th S/C viats from L point b aing th vlocit corrction V at L, an thn S/C rachs som subsqunt prig point whr th scon vlocit corrction V is carri out to scap from th Earth s gravit. Fig. 7 illustrats th initial phas that S/C parts from L at an angl of L θ with V an thn arrivs at som subsqunt prig point at an angl of, whr x-axis is fin as zro gr an th arrow of angl inicats th positiv irction. Lt L an V b th acobi intgral valus at L with S/C initial stat (i.., zro vlocit) an aftr hav bn appli, rspctivl. Not that th S/C position os not var uring th impulsiv manuvr; Eq. (4) ils V L V () Whn Eq. (4) is us, th vlocit at som subsqunt prig point v is xprss as blow. θ

4 µ µ v ( x ) (3) p S. whr with cosin formulation it is foun that x + µ ) + ( µ ) cos( π θ ) ( S. (4) + cos( π θ ) (5) Nxt, Fig. 8 illustrats th nxt stp that is appli in th vlocit irction at som subsqunt prig point to scap from th Earth s gravit. A larg nrg can b fficintl obtain with small V at th nar arth rgion. Lt V b acobi Intgral valu aftr at som subsqunt prig point. Eq. (6) is obtain from Eq. (4): + V V + v ( v ) (6) In analog with Eq. (3), th scap vlocit at th bounar of Earth s gravitational sphr v is obtain v µ µ x V ( x ) (7) S. whr with cosin formula it is foun that x + ( µ ) + ( µ ) cos( π θ) (8) S. + cos( π θ ) (9) C3.0 km /s conition at th bounar of Earth gravitational sphr is obtain b incrasing th S/C vlocit v with an V. Th vlocit variation V is V + (0) From Eqs. (5), (6) an () through (0), it is foun that is trmin b th following factors; V V ) first vlocit corrction ) prig positionθ 3) prig altitu 4) scap vlocit irctionφ 5) scap positionθ In th following, w iscuss th charactristics of twoimpuls scap stratg focusing on th two factors: V an. Th rmaining thr factors wr also invstigat but not shown for brvit. Fig. 9 shows th analticall riv V plott against V. It is foun that V crass with crasing V. It is bcaus that S/C can approach th nar arth rgion vn if V is infinitsimal an th arth scap is attain b prforming small V at that prig point. Not that V 0.74 km/s agrs with th analticall riv V valu using on-impuls scap stratg whn th scap position is 0 g an th scap vlocit irction is -90 g (s Fig. 5), whn th two-impuls scap stratg is gnrat into th onimpuls scap stratg. Fig. 7 Two-impuls scap in th rotating sstm (from L to som subsqunt prig point). Fig. 8 Two-impuls scap in th rotating sstm (from som prig point to gravitational sphr). Fig. 9 Rlation btwn analticall riv V V an (th scap position is 0 g an th scap vlocit irction is -90 g). Fig. 0 shows th rlation btwn analticall riv V an th prig altitu of V. Th prig as wll as th magnitu V at L point ar chang which caus a variation of th analticall riv V. Ths V cras with crasing th prig altitu. It is bcaus that a larg nrg can b fficintl obtain with small V if th prig altitu is low. Morovr, this figur inicats that th valu of

5 V bcom lagr than 0.74 km/s pning on th prig position, whn th prig altitu is highr than on million km. This mans V of two-impuls scap possibl bcoms largr than that of on-impuls scap whn th prig altitu is highr than on million km. Thrfor, it can b shown that two-impuls scap stratg V is ffctiv whn laving L point with infinitsimal an prforming V at a low altitu. Fig. Exampl of th optimal two-impuls scap trajctor in th rotating fram. Fig. 0 Rlation btwn th analticall riv V an th prig altitu (th scap position is 0 g an th scap vlocit irction is -90 g). Optimal Escap Trajctoris of Two-Impuls Escap Stratg In this sction, th scap trajctor of two-impuls scap stratg is also optimiz b SQP in consiration of th following factors. Objctiv function: + min θ Paramtr: partur angl L, magnitu of first an scon vlocit corrction V, V Constraint conition: C3.0 km /s, scap Th optimal rsults of th two-impuls scap trajctor b SQP ar shown in Figs. through 3. Fig. shows th xampl of th optimal two-impuls arth scap trajctor with small V ( 0.08 km/s) an V ( 0.4 km/s) at th fourth prig point whr th prig altitu ( 90,000 km) is th lowst within on-ar flight tim. Fig. shows th analticall riv V b th acobi Intgral an prsnts th optimal minimum rquir V b SQP at onl svral scap positions which ara ar convrg b SQP. W spculat that it can scap comparativl smallr V at ths aras. Fig. 3 shows th rlation btwn rquir V an th scap vlocit irction. Fig. Comparison of th analticall riv with optimal rquir stratg). V Fig. 3 Rlation btwn optimal rquir V (two-impuls scap V an scap vlocit irction (two-impuls scap stratg). Optimal Escap Trajctoris of Multi-Impuls Escap Stratg Morovr, th scap trajctor of multi-impuls scap stratg is optimiz b SQP in consiration of th following factors.

6 Objctiv function: N i min (N an flight tim pn on th scap trajctor shap) Paramtr: partur angl θ L, magnitu of ach vlocit corrction i,,, ) V ( N i Constraint conition: C3.0 km /s, scap i, Vi <.0 km/s ( i,,, N ) Th optimal rsults of th two-impuls scap trajctor b SQP ar shown in Figs. 4 through 7. Fig. 4 shows th xampls of multi-impuls irct scap trajctor (Tp-A) an Fig. 5 shows th rlation btwn V an th flight tim in this cas. It is foun that i ar prform at onl th vicinit of L point, multi-impuls scap stratg is gnrating to onimpuls scap stratg. Fig. 6 shows th xampls of multi-impuls scap trajctor which approachs th arth onc (Tp-B). An Fig. 7 shows th rlation btwn V an th flight tim in this cas. ar prform at th vicinit of L point an th arth, thn it can b sai multi-impuls scap stratg is almost gnrating to two-impuls scap stratg. Morovr, w ar also unr xamination about othr multi-impuls scap trajctor tps now. i Fig. 6 Exampl of th multi-impuls scap trajctor in th rotating fram (Tp-B). Fig. 7 Rlation btwn an Flight tim (Tp-B). Fig. 4 Exampl of th multi-impuls scap trajctor in th rotating fram (Tp-A). Fig. 5 Rlation btwn an Flight tim (Tp-A). Comparison of numrical on-impuls, two-impuls scap stratgis an multi-impuls rsults Th on-impuls, th two-impuls scap stratgis an th multi-impuls scap stratg ar compar in Figs. 8 an 9. Fig. 8 shows that th flight tim of on-impuls scap taks lss than 50 as, an that of two-impuls scap taks from 70 to 360 as, which is longr than that of onimpuls scap bcaus S/C coms clos to th Earth with V onc an scaps from th Earth s gravit. An th flight tim of multi-impuls scap pns on th tp of trajctor. As mntion arlir, tp-a an tp-b ar gnrating to on-impuls an two-impuls scap stratg rspctivl. Thn, th flight tim of tp-a an on-impuls scaps stratg is almost th sam valu in cas of th scap vlocit irction 0 g, an w can spculat th flight tim of tp-b an of two-impuls scaps stratg is almost th sam valu in cas of scap vlocit irction -60 g although th flight tim of two-impuls scaps stratg is not obtain. Fig. 9 shows th comparison of among th onimpuls, th two-impuls an th multi-impuls scap stratgis. Th of multi-impuls scap stratg shoul b minimum valu, on-impuls scap stratg can b prsum th most ffctiv stratg for V whn th scap vlocit irction is aroun 0 gr. Th two-

7 impuls stratg might b also th bst stratg for V whn th scap vlocit irction is aroun 60 gr. Th optimal scap stratg for V might pn on th scap vlocit irction. Fig. 8 Comparison of flight tim among on-impuls, two-impuls an multi-impuls scap stratgis. Fig. 9 Comparison of V among on-impuls, two-impuls an multi-impuls scap stratgis. 5. CONCLUSION Th arth scap trajctoris starting from L of th Sun-Earth sstm insta of th low arth orbit wr invstigat in th planar circular rstrict Sun-Earth-S/C thr-bo problm. Firstl, on-impuls scap an twoimpuls scap wr analz using acobi intgral b fixing th arth scap nrg (C3) to stimat th minimum rquir vlocit corrction. For th twoimpuls cass, it was foun that smallr first vlocit corrction at L follow b th scon vlocit corrction at th low prig altitu rsult in lowr vlocit corrction. Nxt, th optimal numrical calculations wr carri out b SQP for ths two stratgis. Comparing th analtical invstigation using acobi intgral with numrical illustration, ths two rsults ar in goo agrmnt in trms of th stimation of th rquir vlocit corrction. For th on-impuls scap stratg, th flight tim taks lss than 50 as an th rquir vlocit corrction is from 0.9 to.6 km/s. For th twoimpuls scap stratg, th flight tim taks mor than 70 as an th rquir vlocit corrction is from 0.3 to 0.5 km/s. Furthrmor, th rsults of multi-impuls scap stratg ar prsnt. Th multi-impuls scap stratg might gnrat to on-impuls or two-impuls scap stratg, th optimal scap stratg for rquir vlocit corrction can b prsum iffrnt with th scap vlocit irction. Th analsis provis th rquir vlocit corrction an th scap irction from th Earth, givn th arth scap nrg (C3), an can b appli for th sign of th intrplantar transfr trajctor. An xtnsion to th othr tp multi-impuls stratg an th us of Halo orbit insta of L point for th starting conition ar plann for th futur work. REFERENCES Topputo, F., Vasil, M., an Finzi, A., An Approach to th Dsign of Low Enrg Intrplantar Transfrs Exploiting Invariant Manifols of th Rstrict Thr- Bo Problm, AAS/AIAA Spac Flight Mchanics Confrnc, Maui, 8- Fbruar, 004. Koon, W.S., Lo, M.W., Marsn,., an Ross, S.D., Htroclinic Connctions btwn Prioic Orbits an Rsonanc Transitions in Clstial Mchanics, Chaos, Vol. 0, No., un 000, p.p Gomz, G., Koon, W.S., Lo, M.W., Marsn,., Masmont,., an Ross, S.D., Invariant Manifols, th Spatial Thr-Bo Problm an Spac Mission Dsign, Procings of AIAA/AAS Astronamics Spcialist Mting, Qubc Cit, Canaa, August, 00, AAS Ross, S.D., Koon, W.S., Lo, M.W., Marsn,., an Masmont,., Application of Dnamical Sstms Thor to a Vr Low Enrg Transfr, AAS/AIAA Spac Flight Mchanics Confrnc, Maui, 8- Fbruar, Yamato, H., an Spncr, D.B., Trajctor Dsign for Planar Circular an Elliptical Rstrict Thr-Bo Problms with Prturbation, ournal of Guianc, Control an Dnamics, Vol. 7, No. 6, pp , Villac, B., an Schrs, D.., Escaping Trajctoris in th Hill Rstrict Thr Bo Problm an Applications, ournal of Guianc, Control, an Dnamics, Vol. 6, No., p.p. 4-3, Gobtz, F.W., an Doll,.R., A Surv of Impulsiv Trajctoris, AIAA ournal, Vol. 7, No. 5, Ma 969, pp Farquhar, R.W., an Dunham, D. W., Obsrvatoris in Earth Orbit an Bon,. Y. Kono (Kluwr Acamic Pub. 990), p.p Kawaguchi,., an Yoshimura, M., Dp Spac Qua at L an Low Thrust Dpartur/Rturn Flight Stratg, 6 th Intrnational Smposium on Spac Flight Dnamics, 3-7 Dcmbr, Matsumoto, M., an Kawaguchi,., Dp Spac Qua an Solar Voag Era, Th Sixth IAA Intrnational Confrnc on Low-Cost Plantar Missions, Koto, apan, -3 Octobr, 005. Nakamia, M., an Yamakawa, H., Analsis of Earth Escap Trajctoris Starting from Sun-Earth L Point, 6th Workshop on Astronamics an Flight Mchanics, ISAS, - August, 006. Tits, A.L., Zhou,.L., an Lawrnc, C.T., A C Co for Solving (Larg Scal) Constrain Nonlinar (Minimax) Optimization Problms, Gnrating Itrats Satisfing All Inqualit Constraints, Institut for Sstms Rsarch TR-94-6r, Univrsit of Marlan.

TWO REFERENCE japollo LUNAR PARKING - ORBITS / T. P. TIMER. (NASA CR OR rmx OR AD NUMBER) OCTOBER 1965 GODDARD SPACE FLIGHT CENTER

x-543-55-399 * 1 TWO REFERENCE japollo LUNAR PARKING - ORBITS / I - -. -! BY T. P. TIMER,< CFSTI PRICE(S) $ c 4 (PAGES1 (NASA CR OR rmx OR AD NUMBER) 277 I (CATEGORY) ff 653 July 65 OCTOBER 1965,r ; I