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1 Heiliger, J., Hiddink,., Noomen, R., and McInne, C. R. (04) Multiple pacecraft tranfer to un-earth ditant retrograde orbit for Ateroid detection miion. In: International Atronautical Congre 04 (IAC 04), 9 ep - 3 Oct 04, Toronto, Canada. Copyright 04 The Author A copy can be downloaded for peronal non-commercial reearch or tudy, without prior permiion or charge Content mut not be changed in any way or reproduced in any format or medium without the formal permiion of the copyright holder() When referring to thi work, full bibliographic detail mut be given Depoited on: 3 October 04 Enlighten Reearch publication by member of the Univerity of Glagow

2 IAC-4-C..7 OLAR AIL PERIODIC ORBIT IN THE EARTH-MOON THREE-BODY PROBLEM Jeannette Heiliger Advanced pace Concept Laboratory, Univerity of trathclyde, United Kingdom, ander Hiddink Delft Univerity of Technology, the Ron Noomen Delft Univerity of Technology, the Netherland, Colin R. McInne Advanced pace Concept Laboratory, Univerity of trathclyde, United Kingdom, olar ailing ha been propoed for a range of novel pace application, including hovering above the ecliptic for high-latitude obervation of the Earth and monitoring the un from a ub-l poition for pace weather forecating. Thee application, and many other, are all defined in the un-earth three-body problem, while little reearch ha been conducted to invetigate the potential of olar ailing in the Earth-Moon three-body problem. Thi paper therefore aim to find olar ail periodic orbit in the Earth-Moon three-body problem, in particular Lagrange point orbit. By introducing a olar ail acceleration to the Earth-Moon three-body problem, the ytem become nonautonomou and contraint on the orbital period need to be impoed. In thi paper, the problem i olved a a twopoint boundary value problem together with a continuation approach: tarting from a natural Lagrange point orbit. the olar ail acceleration i gradually increaed and the reult for the previou ail performance i ued a an initial gue for a lightly better ail performance. Two in-plane teering law are conidered for the ail, one where the attitude of the ail i fixed in the ynodic reference frame (perpendicular to the Earth-Moon line) and one where the ail alway face the un. The reult of the paper include novel familie of olar ail Lyapunov and Halo orbit around the Earth-Moon L and L Lagrange point, repectively, for both olar ail teering law. Thee orbit are double-revolution orbit that wind around or are off-et with repect to the natural Lagrange point orbit. Finally, the effect of an out-of-plane olar ail acceleration component and the un-ail configuration i invetigated, giving rie to additional familie of olar ail periodic orbit in the Earth-Moon three-body problem. I. INTRODUCTION olar ail technology i rapidly gaining momentum after recent uccee uch a JAXA IKARO miion [] and NAA Nanoail-D miion []. Reearch in the field i flourihing and new olar ail initiative are cheduled for the future, including NAA unjammer miion [3] (launch pot-05) and The Planetary ociety Lightail- miion [4] (launch 06). Additional propoal include NAA Lunar Flahlight and NEA cout [5] miion. The potential of olar ailing lie in the fact that, contrary to other low-thrut propulion technologie like electric propulion, olar ailing doe not rely on an onboard propellant ource. By exploiting the radiation preure generated by olar photon reflecting off a large, highly reflective membrane, olar ail can produce a continuou thrut force that i only limited by the lifetime of the membrane material in the pace environment. They therefore enable long-lived and high-energy miion concept that have abundant novel application [6]. One dynamical ytem commonly ued to decribe the motion of the olar ail i the un-earth circular retricted three-body problem (CR3BP). Thi wellknown ytem yield five natural equilibrium olution (the L to L 5 Lagrange point). Adding a propulive thrut force, uch a generated by a olar ail, to thi CR3BP complement thee five Lagrange point with an infinite et of artificial equilibrium point (AEP). The literature propoe the ue of thee AEP to hover over the pole of the un for heliophyic [7] or to hover along the un-earth line for pace weather forecating [3]. In addition, periodic orbit around thee AEP have been uggeted for further pace weather monitoring capabilitie [8] and parking a pacecraft above the orbit of the Earth for high-latitude obervation, navigation and communication [9]. A dynamical ytem le invetigated for olar ailing i the Earth-Moon CR3BP, the mot likely reaon being that the ytem i non-autonomou: a the un can be conidered to rotate around the Earth-Moon ytem once per ynodic lunar month, the direction of the photon impinging on the olar ail change accordingly. A a reult, AEP in the Earth-Moon problem are only tationary intantaneouly: their coordinate are time-dependent and either control i IAC-4-C..7 Page of

3 required to compenate for the moving un-line or the ail ha to navigate along the time-changing AEP [0]. Finding olar ail periodic orbit in the Earth-Moon ytem add another complexity in that the period of the orbit ha to be equal to a fraction (or multiple) of the ynodic lunar month in order for the orbit to be periodic. Previou work on olar ail periodic orbit in the Earth-Moon ytem either linearied the equation of motion [, ] or earched for bepoke orbit (e.g. below the lunar outh Pole [3]) by olving the optimal control problem. Thi work will intead look for entire familie of olar ail periodic orbit, in particular Lyapunov and Halo orbit, in the Earth-Moon ytem by olving the accompanying boundary value problem. The exitence of uch familie, cloer to Earth than poible in the un-earth ytem, can potentially give rie to unrivalled Earth obervation capabilitie or lunar faride communication capabilitie. The tructure of the paper i a follow. Firt, the non-autonomou dynamical ytem i decribed in ection II, followed by a definition of the two-point boundary value problem (BVP) to be olved in ection III. ection IV ubequently decribe the initial gue and continuation cheme ued to olve the twopoint BVP for increaing olar ail performance. Then, after decribing the adopted in-plane olar ail teering law in ection V, the reult are preented in ection VI and VII for olar ail Lyapunov orbit at the L -point and olar ail Halo orbit at the L -point, repectively. Then, in ection VIII further contraint on the olar ail teering law are invetigated, while ection IX conider an out-of-plane olar ail acceleration component. Finally, the effect of the un-ail configuration at time t 0 (i.e. the choice for the initial condition) i invetigated in ection X and the paper end with the concluion. II. DYNAMICAL YTEM In the circular retricted three-body problem (CR3BP), the motion of an infinitely mall ma, m, (i.e. the olar ail pacecraft), i decribed under the influence of the gravitational attraction of two much larger primary mae, m (the Earth) and m (the Moon). The gravitational influence of the mall ma on the larger mae i neglected and the larger mae are aumed to move in circular orbit about their common center-of-ma. Figure how the reference frame that i employed in the CR3BP: the origin coincide with the center-ofma of the ytem, the x -axi connect the larger mae and point in the direction of the maller of the two, m, while the z -axi i directed perpendicular to the plane in which the two larger mae move. The y - axi complete the right handed reference frame. Finally, the frame rotate at contant angular velocity,, about the z -axi, ω z. ˆ New unit are introduced (ee alo Table ): the um of the two larger mae i taken a the unit of ma, i.e. m m. Then, with the ma ratio m / m m, the mae of the large bodie become m and m. A unit of length, the ditance between the main bodie i elected, and / i choen a unit of time yielding, and o one lunar month i repreented by. The converion factor to change between dimenional and nondimenional mae, ditance and time are given in Table. z ω m Figure chematic of CR3BP Figure chematic of non-autonomou Earth-Moon three-body problem Earth-Moon CR3BP Total ma ( m m) kg Earth-Moon ditance 384,40 km 5 One time unit O t Earth Table Earth-Moon CR3BP parameter y r r y Ŝ ail r m a Moon x x IAC-4-C..7 Page of

4 In thi reference ytem, the motion of the olar ail i decribed by: [6] with T r ωr ω ωr a V () r x y z the poition vector of m. The term on the left hand ide are the kinematic, corioli and centripetal acceleration, repectively, while the term on the right hand ide are the olar ail acceleration and the gravitational acceleration exerted by the primary mae. The gravitational potential, V, i given by: V () r r with the vector r and r defined a T r x y z and r x y z. Following Reference [6], the centripetal acceleration in Eq. () can be written a the gradient of a calar potential function, ωr, and can be combined with the gravitational potential into a new, effective potential, U : x y U r r The new et of equation of motion then become: T (3) r ωr a U (4) Finally, for the olar ail acceleration, an ideal ail model i aumed [6]. An ideal olar ail i a ail that i perfectly reflecting. The incoming olar photon are therefore pecularly reflected and the olar radiation preure force act perpendicular to the ail urface, in direction ˆn. Furthermore auming that the olar radiation preure i contant in magnitude throughout the Earth-Moon ytem, the olar ail acceleration, a, can be written a: 0, EM ˆ ˆ a a n n ˆ (5) In Eq. (5), Ŝ i the direction of the un-line, ee Figure, which can be expreed a: t t ˆ co in 0 T (6) with the angular rate of the un-line in nondimenional unit, ee alo Table. Note that Eq. (6) ignore the mall inclination difference between the un-earth and Earth-Moon orbital plane and at time t 0, the un i aumed to be on the negative x -axi. Finally, note that a i the ail characteritic acceleration in non-dimenional unit. The characteritic acceleration i the acceleration generated by the olar ail when it face the un at Earth ditance (at Atronomical Unit). Derived from the unjammer ail performance, a typical characteritic i 0.5 mm/ [4], which tranlate into a value for a of III. BOUNDARY VALUE PROBLEM Due to the non-autonomou behaviour of the ytem, periodic orbit are found by treating the problem a a two-point boundary value problem (BVP) rather than uing conventional method uch a differential corrector [5]. The dynamic of the BVP are given in Eq. (4) and the boundary contraint are given by: t0 tf x x (7) with x r r T, the indice 0 and f indicating the initial and final condition and t, i.e. one ynodic lunar month. The BVP i olved uing the collocation method implemented in the MATLAB bvp4c.m function. IV. INITIAL GUE To olve the two-point boundary value problem, bvp4c.m need an initial gue. Here, a continuation approach i adopted, where the earch for olar ail periodic orbit tart from a natural periodic orbit with a uitable period. The ail performance in term of characteritic acceleration a i then lowly increaed and the olution for the previou value for a i ued a initial gue for the next value for a. Thi will in the end give rie to familie of periodic orbit for increaing value of the ail performance. IV.I Lyapunov orbit around L The election of the initial, natural Lyapunov orbit around the L Lagrange point i hown in Figure 3a. Thee natural Lyapunov orbit are generated uing the approach decribed in Reference [5] and their orbital period are provided in Figure 3b. Only the Lyapunov orbit with a period equal to a fraction of the ynodic lunar month would be uitable initial gue candidate, i.e. the following contraint applie: P Nat Lyap with i,,3... i The orbit that fulfil thi contraint i indicated with a thick blue line in plot and with a blue aterik in plot. The elected Lyapunov orbit ha a period of P and thu make two orbital revolution in Nat Lyap one ynodic lunar month. f (8) IAC-4-C..7 Page 3 of

5 Figure 3 Lyapunov family at L. Lyapunov orbit. Period of Lyapunov orbit in. Figure 4 Northern Halo family at L. Halo orbit. Period of Halo orbit in. IV.II Halo orbit around L The approach to find the initial, natural Halo orbit around the L Lagrange point i very imilar to the approach decribed for the Lyapunov orbit around L. The family of natural (northern) Halo orbit around L i provided in Figure 4a with their orbital period in Figure 4b. Again, the orbit with a period equal to a fraction of the ynodic lunar month i choen a uitable initial gue, which in thi cae alo ha a period of P and therefore make two orbital Nat Halo revolution in one ynodic lunar month. V. OLAR AIL TEERING LAW Different familie of olar ail periodic orbit can be generated for different olar ail teering law. The two law invetigated in thi paper are: ) The ail normal i alway directed along the Earth- Moon line, i.e. along the x -axi: n ˆ ign co 0 0 T (9) Thi teering law allow for a contant attitude of the ail in the CR3BP reference frame (and therefore a contant acceleration direction), but implie a changing olar ail acceleration magnitude. Note that the term ign co take into account that the olar ail acceleration change ign when the un move from a poition where it illuminate the Earth-facing ide of the ail to a poition where it illuminate the Moon-facing ide of the ail. Note that thi implie that the olar ail ha to be reflective on both ide. ince, conventionally, olar ail are only reflective on one ide (to allow thermal emiion on the rear ide), ection VIII will conider the effect of contraining the teering law for a one-ided reflective olar ail. ) The ail normal i alway directed along the unail line, i.e. the ail alway face the un: nˆ ˆ (0) Contrary to the firt teering law, thi teering law allow for a contant magnitude of the ail acceleration a a ), ˆ but require a changing attitude of the ( 0, EM ail in the CR3BP reference frame. IAC-4-C..7 Page 4 of

6 VI. REULT L LYAPUNOV ORBIT Thi ection preent the familie of olar ail Lyapunov orbit around the Earth-Moon L -point and for the two teering law outlined in the previou ection. The maximum olar ail characteritic acceleration conidered i a 0.088, which i lightly larger than unjammer performance, but indicate what could be feaible in the near-term. VI.I ail normal along Earth-Moon line When auming that the ail normal i alway directed along the Earth-Moon line, the family of olar ail Lyapunov orbit a hown in Figure 5a can be obtained: the family originate from the natural Lyapunov orbit (ee Figure 3) and i created by lowly increaing the characteritic acceleration, a, of the olar ail. The larger the value for a, the more the olar ail orbit deviate from the natural Lyapunov orbit. The orbit for the larget characteritic acceleration, i.e. a = 0.088, i provided in Figure 5b which clearly how the difference between the natural and olar ail orbit: the olar ail Lyapunov orbit wind around the natural orbit and clearly make two revolution within one ynodic lunar month. A the ail can only produce the maximum olar ail acceleration at time t 0 and t (and generate no acceleration at t and t 3 ), the offet between the natural and olar ail orbit i only mall. Note that the aterik indicate the initial condition of the orbit, which i choen to be the y -axi croing on the Earth-ide of the L -point. ection X will invetigate the influence of chooing a different initial condition on the hape of the olar ail periodic orbit. VI.II ail normal along un-ail line The family of olar ail Lyapunov orbit around the L -point that can be generated by orienting the ail normal along the un-ail line i provided in Figure 6a. The change in orbit compared to the natural Lyapunov orbit i much greater than for the Earth-Moon line teering law in Figure 5 a the maximum ail acceleration can be exploited throughout the entire orbital period. Figure 5 olar ail Lyapunov orbit at L with Earth- Moon line teering law. The thick cyan orbit i the natural Halo orbit in Figure 3a. Family of orbit. Orbit for a (aterik i initial condition) Figure 6 olar ail Lyapunov orbit at L with un-ail line teering law. The thick cyan orbit i the natural Lyapunov orbit in Figure 3a. Family of orbit. Orbit for a (aterik i initial condition). IAC-4-C..7 Page 5 of

7 The greater difference between the natural and olar ail orbit i alo clear from Figure 6b that how the olar ail Halo orbit for a = Thi orbit clearly how the two orbital revolution per ynodic lunar month with an inner and outer loop that are much maller and greater than the natural Lyapunov orbit, repectively. VII. REULT L HALO ORBIT Thi ection preent the familie of olar ail Halo orbit around the L -point of the Earth-Moon ytem and for the two teering law a outlined in ection V. VII.I ail normal along Earth-Moon line When auming that the ail normal i alway directed along the Earth-Moon line, the family of olar ail Halo orbit a hown in Figure 7 can be obtained. When comparing with the reult for the Lyapunov orbit in Figure 5 (i.e. uing the ame olar ail teering law), a very imilar effect of the olar ail acceleration on the in-plane motion can be oberved: the olar ail periodic orbit wind around the natural periodic orbit. However, for the Halo orbit in Figure 7, the olar ail acceleration alo affect the out-of-plane motion a it flatten the Halo orbit, poitioning the outhern part of the orbit cloer to the Moon. From Figure 8, which how the olar ail orbit for the larget characteritic acceleration conidered, thi i particularly clear. Again, the aterik in plot c-d indicate the initial condition, which i choen to be the y -axi croing on the faride of the L -point. Figure 8 olar ail Halo orbit at L with Earth-Moon line teering law and a The thick cyan orbit i the natural Halo orbit in Figure 4a (aterik i initial condition). VII.II ail normal along un-ail line The final family to be conidered i the olar ail Halo orbit family at the L -point with a un-ail line teering law. The reult are preented in Figure 9, while Figure 0 provide the orbit for two characteritic acceleration, a = (plot a- and a = (plot c-d). Figure 9 how a very intereting reult: by increaing the olar ail performance, the out-of-plane motion of the olar ail Halo orbit decreae until the Halo orbit reduce to a planar olar ail Lyapunov orbit. A imilar effect can potentially be expected when further increaing the ail performance for the Earth- Moon line teering law in ection VII.I. Figure 7 Family of olar ail Halo orbit at L with Earth-Moon line teering law. The thick cyan orbit i the natural Halo orbit in Figure 4a. IAC-4-C..7 Page 6 of

8 Figure 9 Family of olar ail Halo orbit at L with unail line teering law. The thick cyan orbit i the natural Halo orbit in Figure 4a. VIII. ONE-IDED REFLECTIVE OLAR AIL A mentioned in ection V, the olar ail teering law in Eq. (9) aume that the olar ail i reflective on both ide. While thi may be achievable in the longterm, a more near-term cenario would be that the olar ail i only reflective on one ide, the front ide (i.e. the Earth-facing ide), and will have a highly thermally emitting rear urface. Thi i to emit the aborbed energy of the mall fraction of incident olar radiation that, in reality, will be aborbed by the ail ubtrate [6]. The rear urface hould therefore not be expoed to unlight. To take thi into account, thi ection impoe an additional contraint that witche the ail off when the un illuminate the rear ide of the ail: 0, EM ˆ ˆ ˆ ˆ ˆ ˆ ˆ a a n n if n 0 a 0 if n 0 () Note that a zero acceleration can be achieved by poitioning the ail edge-wie to the un-line. Alo note n ˆ ˆ co t and therefore the contraint that a 0 i impoed when 3 t. c) d) Figure 0 olar ail Halo orbit at L with un-ail line teering law. The thick cyan orbit i the natural Halo orbit in Figure 4a. The aterik are the initial condition. a a IAC-4-C..7 Page 7 of

9 The reult i provided in Figure and Figure for the olar ail Lyapunov orbit at L and the olar ail Halo orbit at L, repectively. When conidering the Lyapunov orbit, it become clear that exploiting the olar ail acceleration only when the un i on the Earth-ide of the orbit reult in very minor deviation from the natural Lyapunov orbit. However, it i clear that thoe part of the orbit where the olar ail acceleration act (i.e. the firt half of the firt revolution and the econd half of the econd revolution) are diplaced toward the un. Conidering the olar ail Halo orbit with the contrained Earth-Moon line teering law, the olar ail acceleration ha a much greater effect even though it only act during half of the orbital period. In the out-ofplane direction a imilar effect a for the orbit in Figure 9 can be oberved, i.e. the out-of-plane motion decreae for increaing ail performance until the orbit reduce to a planar olar ail Lyapunov orbit. c) Figure olar ail Halo orbit at L with contrained Earth-Moon line teering law. The thick cyan orbit i the natural Halo orbit in Figure 4a. a- Family of orbit. c) Orbit for a (aterik i initial condition). Figure olar ail Lyapunov orbit at L with contrained Earth-Moon line teering law. The thick cyan orbit i the natural Halo orbit in Figure 3a. Family of orbit. Orbit for a (aterik i initial condition). IX. OUT-OF-PLANE OLAR AIL ACCELERATION The teering law propoed in ection V only conider an in-plane olar ail acceleration. However, the olar ail can be pitched with repect to the Earth- Moon plane to alo create an out-of-plane acceleration component. Thi ection will demontrate the potential of thi out-of-plane teering law by uing the olar ail Lyapunov orbit at the L -point with a un-ail line teering law a a tet cae. Then, rather than adopting the in-plane law, nˆ, ˆ the following out-of-plane unail line teering law i conidered, ee alo Figure 5: IAC-4-C..7 Page 8 of

10 65th International Atronautical Congre, Toronto, Canada. Copyright 04 by Jeannette Heiliger. Publihed by the IAF, with permiion and n co co t co in t in () T Note that for a pitch angle = 0, Eq. () reduce to the in-plane teering law. pitch angle i lowly increaed, the BVP i olved and the reult i ued a an initial gue for a lightly larger value for. The family of orbit that reult from thi i provided in Figure 3a-b for a range of pitch angle, 60,60 in deg. Note that the orbit for negative and poitive value for the pitch angle are the ame, only mirrored in the x, y -plane, and create out-ofplane diplacement above en below the x, y -plane, repectively. The maximum out-of-plane diplacement that i achieved in each orbit of the family i provided in Figure 4, which clearly how that for a pitch angle of 34.5 deg, the orbit extend farthet above or below the Earth-Moon plane. The correponding orbit i depicted in Figure 3c and reache an out-of-plane diplacement of approximately 0,50 km. c) Figure 4 Maximum out-of-plane diplacement in each orbit of the family of olar ail Lyapunov orbit at L with out-of-plane un-ail line teering law (Figure 3) a a function of the olar ail pitch angle,. z n y Figure 3 olar ail Lyapunov orbit at L with out-ofplane un-ail line teering law and a Orbit in blue and red are for > 0 and < 0, repectively. The thick cyan orbit i the natural Halo orbit in Figure 4. a- Family of orbit projected on the (x,y)- and (x,z)-plane. c) Orbit with maximum out-of-plane diplacement ( = 34.5 deg) (aterik i initial condition). To generate the family of out-of-plane un-ail line Lyapunov orbit at L, a continuation in i applied: tarting from the planar olar ail Lyapunov orbit with = 0 and a in Figure 6b, the value for the IAC-4-C..7 Earth-Moon plane x Figure 5 chematic of out-of-plane un-ail line teering law X. INITIAL CONDITION AT TIME t = 0 A indicated in ection II, the poition of the un at time t 0 i aumed to be on the negative x -axi. The actual un-ail configuration throughout the orbit and during one ynodic lunar month then depend on the choice for the initial condition along the natural Halo orbit from where the continuation for olar ail periodic orbit tart. The choice for thi initial condition will Page 9 of

11 influence the hape of the familie of olar ail periodic orbit, which i demontrated in thi ection by invetigating different initial condition for the olar ail Halo orbit with a contrained Earth-Moon line teering law, ee ection VIII. A mentioned in ection VII, the reult of ection VII.I are obtained by chooing the initial condition to be the y -axi croing on the faride of the L -point, i.e. below the Earth-Moon plane. A alternative, thi ection conider the y -axi croing on the Earth/Moon-ide of the L -point, i.e. above the Earth-Moon plane. When doing o, the reult in Figure 6 i obtained. Comparing Figure and Figure 6 immediately how the effect of the choice for the initial condition: when chooing the Earth-ide y -axi croing a initial condition, the olar ail Halo orbit increae in ize compared to the natural Halo orbit, while for chooing the far-ide y -axi croing a initial condition, the olar ail Halo orbit decreae in ize. Thi implie that very trict inertion contraint need to be impoed in order for the olar ail to enter the deired orbit. Note that the effect on the out-of-plane motion i very imilar for both initial condition, i.e. the Halo orbit eventually reduce to planar Lyapunov orbit. CONCLUION Thi paper ha demontrated the exitence of olar ail periodic orbit in the non-autonomou Earth-Moon ytem. By olving the accompanying two-point boundary value problem and uing a continuation approach, entire familie of olar ail L -Lyapunov and L -Halo orbit have been found for increaing olar ail performance. Due to the non-autonomou behaviour of the ytem, all orbit have a period of one ynodic lunar month and make two revolution per orbital period. In addition, different familie have been obtained for different in-plane olar ail teering law: either keeping the olar ail perpendicular to the Earth-Moon line or to the un-ail line. A the latter can continuouly exploit the maximum achievable olar ail acceleration, the familie with a un-ail line teering law how a greater difference with repect to the natural Lyapunov or Halo orbit from which they bifurcate than the Earth-Moon line teering law. For both law, the in-plane motion of the olar ail periodic orbit wind around the natural Lyapunov/Halo orbit, while for the Halo orbit the olar ail caue an off-et in the out-of-plane motion toward the Moon. When increaing the olar ail performance far enough, the Halo orbit even reduce to planar Lyapunov orbit. Furthermore, by introducing an out-ofplane teering law, the family of olar ail Lyapunov orbit at L could be extended in the out-of-plane direction, achieving the maximum diplacement for a ail pitch angle of 34.5 deg. Finally, the effect of different initial condition, i.e. different initial un-ail configuration, have been invetigated, howing a ignificant effect on the olar ail Halo orbit at L. Thi implie that inertion condition are very trict in order to enure that the deired olar ail Halo orbit i achieved. c) Figure 6 Effect of choice for initial condition at time t 0 on the olar ail Halo family at L for an Earth- Moon line teering law and for a The blue olid line i the olar ail Halo orbit for r 0 coinciding with the y -axi croing on the far-ide of the L -point (blue aterik), the red dahed line i the olar ail Halo orbit for r 0 coinciding with the y -axi croing on the Earth/Moon-ide of the L -point. ACKNOWLEDGEMENT Thi work wa funded by the European Reearch Council Advanced Invetigator Grant-757: Viionary pace ytem: Orbital Dynamic at Extreme of pacecraft Length-cale. IAC-4-C..7 Page 0 of

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