INDIRECT PLANETARY CAPTURE VIA PERIODIC ORBITS ABOUT LIBRATION POINTS

Transcription

1 INDIRECT PLANETARY CAPTURE VIA PERIODIC ORBITS ABOUT LIBRATION POINTS Li Xiangyu 1,2, Qiao Dong 1,2, Cui Pingyuan 1,2 (1. Institute of Dee Sace Exloration Technology, Beijing Institute of Technology, Beijing, China, ; 2. Key Laboratory of Autonomous Navigation and Control for Dee Sace Exloration, Ministry of Industry and Information Technology, Beijing, China, ) ABSTRACT The libration oints and their eriodic orbits ossess unique dynamics roerties in multi-body system, which have been exloited to design low-energy transfer in sace missions. In this aer, we investigate the eriodic orbits for lanetary cature and roose an indirect lanetary cature method. The eriodic orbit is used as a ark orbit during the cature, which connects with the interlanetary trajectory and the mission orbit by stable and unstable manifolds, resectively. The arking orbit selection is investigated under Sun-Mars system. The candidate lanar and satial orbits are obtained. The simulation shows that comared with direct cature the indirect cature via eriodic orbits could save velocity increment. Better efficiency can be found for high altitude and high orbit cature. This cature strategy may be of interest for future exloration missions because of low cature velocity, flexibility of transfer and extra scientific returns. Index Terms Planetary cature, eriodic orbit, invariant manifolds 1. INTRODUCTION The lanetary cature is a key rocess for exloration mission, which transfers the sacecraft from interlanetary trajectories to mission orbits around lanets. The cature trajectory strategy lays an imortant role in trajectory design. It not only affects the sacecraft design but also relate to the mission lanning. So far, several cature strategies have been roosed and alied to lanetary cature, including direct cature, aerocature and ballistic cature. The direct cature is the traditional cature method, which directly inserts the sacecraft into target orbit by erforming single imulse maneuver. Direct cature is easy to design but the cost is generally large. The aero-brake technique is another kind of cature method, which takes advantage of the aerodynamic force to reduce the velocity of sacecraft or change the orbital arameters. It has been successfully alied to Mars exloration [1-2]. Though aero-braking shows high efficiency, it roduces high temerature and overload during the cature rocess, which leads to high requirements for sacecraft design. With the understanding of multi-body dynamics and the definition of weak stability boundaries and stable sets, another cature method, so-called ballistic cature has been roosed [3-]. The ballistic cature exloits the gravitational force of lanets to cature a sacecraft. Several lunar missions have used ballistic cature, showing good erformance in fuel consumtion [6-7]. However, it is found that ballistic cature for lanetary cature is hard to achieve, if the excess velocity is high [8-11]. The libration oints, eriodic orbits and their associated manifolds are also unique dynamics roerties in multi-body system. They have been exloited to design low-energy lunar transfer [12], interlanetary transfer [13] and transfer between different orbits [14]. The cature trajectories to eriodic orbits are also studied. Nakamiya and Scheeres et al. investigated the cature to eriodic orbit using imulsive maneuvers at eriais of manifold [1] and aly to Earth-Mars transfer [16]. Wang rovided more cature oortunity by erforming extra more imulse maneuver [17]. However, cature the sacecraft into lanetary orbits via eriodic orbits has not been further studies. In this aer, based on the work of Nakamiya, Scheeres and Wang, we extend the eriodic cature to lanetary orbit cature. The indirect lanetary cature via libration oints and eriodic orbits is roosed and the design rocess is given. The sacecraft is firstly catured into eriodic orbit using stable manifolds and then is transferred into the mission orbit by unstable manifolds. The indirect cature method is researched under the background of Mars. Firstly, the dynamic model of CRTBP in Sun-Mars system is established. Secondly, the rocess for indirect cature is described. The candidate arking orbits are obtained in lanar and satial conditions. Finally, the efficiency of indirect cature method is evaluated and comared with other cature methods. The results show that indirect cature method requires less velocity increment than direct cature. Better efficiency can be found for high altitude and high orbit cature. It also shows advantages in flexibility of transfer, extra scientific returns. The indirect

2 lanetary cature can also be alied to other lanets and rovide reference for future exloration mission. 2. EQUATIONS OF DYNAMICS In this aer, the circular restrict three body model are used, which describes the motion of a massless article under the gravity attraction of two rimaries moving on a circular orbit around their common center of mass. Here, the two rimaries are the Sun ( ) and Mars ( ), and the article is the sacecraft. The motion is studied under the rotating reference frame, in which the origin is located at the barycenter of system The X-axis oints from to, the Z axis aligns with the direction of angular momentum of the system and Y axis comlete the coordinate frame. In the CRTBP, the non-dimensional equations of motion for the sacecraft can be written as, (1 )( x ) ( x1 ) x 2y x 3 3 r1 r2 (1 ) y y y 2x y 3 (1) 3 r1 r2 (1 ) z z z 3 3 r1 r2 where m1 ( m1 m2 ) is the mass arameter of the system, and are the distances from the sacecraft to Sun and the Mars, resectively. Five libration oints exist in the CRTBP, three of them are collinear libration oints (namely L1, L2, L3) and the other two are triangular libration oints (namely L4, L). Sacecraft on those oints kee balance under the mutual influence of rimaries. In this aer, we ay attention to collinear equilibria L1 and L2. There are several kinds of eriodic orbits around L1 and L2, including lanar Lyaunov orbits, vertical Lyaunov orbits and halo orbits. The stability of eriodic orbit could be obtained by the Monodromy of orbits, which defined as the state transition matrix of orbits in one orbital eriod. If any eigenvalue of Monodromy larger than 1, the orbit is unstable. The direction of corresonding eigenvector is noted as unstable direction. According to former research, all eriodic orbits around L1/L2 are unstable. Small erturbations along unstable direction cause the sacecraft asymtotically deart from eriodic orbits. The set of those trajectories are known as unstable invariant manifolds. On the other hand, the eriod orbit also ossesses stable direction, and stable invariant manifolds. Sacecraft on those trajectories will asymtotically arrive onto the eriodic orbit. In this aer, both stable and unstable invariant manifolds are emloyed during the cature rocess. r 1 r 2 m 1 m 2 m 1 m 2 3. CONCEPT OF INDIRECT PLANETARY CAPTURE The concet of indirect lanetary cature is as follow, (1) The first imulse is erformed at the eriais of interlanetary trajectory to inject the sacecraft into stable manifold of eriodic orbits. Then the sacecraft could asymtotically arrive onto eriodic orbits. (2) After one or several eriods on the arking orbit, the sacecraft will deart from the eriodic orbit and aroach the Mars along unstable manifolds. (3) When the unstable manifold arrives its eriais, with the third imulsive maneuver, sacecraft is catured into the mission orbit. The indirect lanetary cature can be searated to three atches, the eriodic arking orbit and its corresonding stable and unstable manifolds. The stable manifolds should connect with hyerbolic trajectory and the unstable manifolds should intersect with mission orbits. The indirect cature usually involves three imulsive maneuvers. The first maneuver and the third maneuver haen at the corresonding eriaides. The magnitude of two maneuvers deends on the excess velocity and the arameters of mission orbits. The second maneuver is usually small but necessary to generate the unstable manifold with secific arameters. One extra maneuver is otional if the erturbed stable manifolds are chosen. The rocess of indirect lanetary cature is illustrated in Fig. 1. Fig. 1 Indirect lanetary cature via eriodic orbit In order to utilized the indirect lanetary cature, both the back and forward time integration are used. Once the arameter of mission orbit is determined, the design rocess could describe as follow, Ste 1. Construct the eriodic arking orbit.

3 Ste 2. Comute the unstable manifolds of the eriodic orbit. The articular stable manifold, which ossess a similar eriais condition as the mission orbit, such as eriais distance, are chosen as the initial value. A differential correction rocess is used to obtained the accurate solution. The second and third imulsive maneuver could be calculated. Ste 3. Comute the stable manifolds of the eriodic orbit to its eriais and record the eriais state. The first imulsive maneuver can be obtained. v, v 2 3 v 1 4. SELECTION FOR PERIODIC ORBITS As mentioned above, the selection of eriodic orbits is a key ste in cature trajectory design, which determines the eriais state of invariant manifolds, including the eriais distance and eriais velocity. There are two criteria to select the eriodic orbits. (1) Energy constrain: The insertion velocity to the stable manifold should be as low as ossible. If we assume the eriais distance of stable manifold is r, the velocity at eriais could aroximately reresent as the escae velocity at such distance ves 2 r. The velocity of the hyerbolic trajectory with excess velocity at eriais is, 2 vex v 2 r.the cost for the first maneuver is aroximately, v1 vex v vex ves v (2) r r The derivate of Eq. (2) to r v is exressed as, 2 v1 1 r r r 2 2 r 2r 2 2 2r v v r r It is clear to show that v v1 r 2 2 (3) 0, 0. That means the first imulsive maneuver v1 increases as the eriais distance increases. The result shows that stable manifold at low eriais requires less cost. Therefore, the eriodic orbits which have low manifold eriais close to the surface of Mars are riority selections. (2) State constrain: The eriais distance of natural unstable manifolds should close to that of mission orbit. Otherwise, the sacecraft need large erturbation maneuver v 2 to correct the eriais distance. Based on the two criteria, suitable eriodic orbits for indirect lanetary cature are investigated in the following arts. 1. Selection in lanar orbits The lanar Lyaunov orbit is firstly investigated. Here we use the amlitude to exress the orbits. Based on A y numerical comutation. The L1 Lyaunov orbits from 4 amlitude A km to km and L2 y Lyaunov orbits from amlitude km are roduced. 400 oints are chosen equally along each eriodic orbits. The stable manifolds are generated from each oints and integrated to eriais backwardly. The eriais distance for each manifold is recorded. According to the analysis above, low eriais distances are referred. Therefore, the eriaides close to the surface of the Mars are shown. Fig. 2 and 3 illustrated the corresondence between eriais distances and orbit amlitude for L1 and L2 oints, resectively. Fig. 2 Corresondence between r km and A yl 1 Fig. 3 Corresondence between r and A yl 2 It is clear to show that not all Lyaunov orbits are suitable for arking orbits. Stable manifolds for small size to

4 orbits do not have low eriais distances, both for L1 and L2 orbits. With the increase of amlitude, the eriaides of stable manifolds are gradually close to the Mars. There is a critical amlitude, in which some branches of natural stable manifold could aroach the Mars in less than 200km. For L2 Lyaunov orbits, the critical amlitude is about A.7 10 km. The critical amlitude for L1 orbit is a little smaller, yc A yc c.. 10 km Besides the eriais distance, the eriais hase angle is another imortant arameter for cature, which determines the aroach direction of hyerbolic trajectory. Here we defined the hase angle as the angle between X-Axis and Mars-Periais line. The hase angle versus eriais distance for L1 and L2 Lyaunov orbits are shown in Fig. 4 and, resectively. A y extend to 140 and 260, which means the eriais of large amlitude orbits have more extensive distribution in the vicinity of Mars. That could rovide more choice for interlanetary transfer. The eriais distances of unstable manifolds are also discussed with different amlitudes. The results for L1 and L2 are shown in Fig. 6 and 7, resectively. As illustrated, the orbits with amlitude larger than A could cover the eriais distance from low Mars orbit ( 389km ) to very high orbit ( km ), which meets requirement for most of mission orbits. Therefore, the constrains of final mission orbits would not affect the arking orbit selection. It only determines the initial outbound osition on arking orbits. yc Fig. 6 Corresondence between ru 1 and A yl Fig. 4 Relation between L1 and r Fig. Relation between L2 and r L1 and L2 Lyaunov orbits ossess totally different eriais hase angles. The hase angles for L1 orbits begin at 10 and extend to 20 and 4 for large amlitudes. The hase angles for L2 orbits distribute in the vicinity of 190. Increasing the orbit amlitude, the hase angles Fig. 7 Corresondence between ru and A yl 2 Based on the analysis above, the candidate arking orbits for indirect cature are from A.10 km to 7.10 km.710 km to for L1 Lyaunov orbit and from km for L2 Lyaunov orbit and. Further selection will decide by final state of interlanetary trajectories. y

5 2. Selection in satial orbits The satial arking orbits are also investigated. Two kinds of eriodic orbit about libration oints are studied. The vertical Lyaunov orbits and halo orbits. For vertical Lyaunov orbits, due to its large amlitude in Z axis, it is found the stable manifolds cannot close to Mars for both L1 and L2 oints. Hence, vertical Lyaunov orbits cannot be alied for indirect cature. Here, we ay attention to discuss the halo orbits. The halo orbits are created based on three-stage analytical and numerical correction. The L1 north Halo 4 4 orbits with amlitude Az km to Az km 4 and L2 south Halo orbits with amlitude Az km to Az 6.10 km are studied. The same roduces above are followed to investigate the candidate arking orbits. The eriais distances of stable manifolds are shown in Fig. 8 for L1 halo orbits and Fig. 9 for L2 halo orbits. As shown in Fig. 8 and 9, the halo orbits from two libration oints have similar critical amlitude Azc km. The eriais states are also investigated in satial situation. The traditional Kelerian arameter cannot resent the osition of eriais clearly. Therefore, we use the eriais hase angle and the satial angel to describe the eriais. The eriais hase angle is modified as the angle between X axis and the rojector of Mars-eriais line in XY lane. The satial angle reresents the angle between Mars-eriais line and XY lane. Another inclination angle is used to describe the velocity direction at eriais. Three angles are shown in Fig. 10. i Fig. 8 Corresondence between r 1 and A zl Fig. 10 Periais state angles The eriais states for L1 stable manifolds is shown in Fig. 11. The eriais for L1 north halo orbit all locate below the XY lane. With the increase of amlitude A z, the eriais of manifold gradually aart from the XY lane. The absolute value of satial angle increases from about at Az km to more than at Az km. The orbital inclination also increases with amlitude. Similar to the lanar situation, the hase angles of manifold still near 10. But the angles barely change as the size of Halo orbit increases. If we substitute the north family to south family. The hase angle 17 A z 40 Fig. 9 Corresondence between r and A zl2

6 (a) (a) (b) (b) (c) Fig.11 the eriais state for L1 stable manifolds (a) orbital inclination angle (b) eriais hase angle (c) eriais satial angle. and inclination do not changes. The eriaides of manifolds locate above the XY lane with the same value of satial angles. The eriais states for L2 stable manifolds is shown in Fig. 12. Periaides for L2 stable manifolds locate in totally different area. With the increase of amlitude, the orbital inclination and satial angle both increase. (c) Fig.12 the eriais state for L2 stable manifolds (a) orbital inclination angle (b) eriais hase angle (c) eriais satial angle. Similar to lanar situation. All candidate halo orbits could satisfy the terminal state constrains. Therefore, the candidate Halo orbits for indirect cature are from Az km and Az km for L1 oints and A km to A km for L2 oints. z z

7 . SIMULATION RESULTS AND DISCUSSION In this art, the efficiency of indirect cature strategy is evaluated under different conditions and comared with direct cature. The interlanetary trajectory is simlified as hyerbolic trajectory with different excess velocity and different tyes of mission orbits are considered. For direct cature method, eriais altitude of hyerbolic trajectory is chosen directly as the eriais of mission orbits. The velocity increment is easy to exress as, where a r a(1 e) 2 2 (1 e) vd v (4) r r is the eriais distance of mission orbit. is the semi-axis of mission orbit and e is the eccentricity of mission orbit. For indirect cature method, the eriais of hyerbolic trajectory is set equal to that of stable manifold. The eriais altitude is not strictly restricted to 200km. The natural stable manifold which has eriais altitude close to 200km are selected without further correction. The first maneuver is v 1 2 () 2 v1 v v r On the other hand, the natural unstable manifold which has the eriais distance close to mission orbit is chosen as the initial guess. A differential correction rocess is used to correct the distance error. The erturbation velocity is noted as. v 2 v 3 Finally, the third velocity increment is imlement at the eriais of unstable manifolds. It can be written as, (1 e) v3 vu (6) r where the velocity of unstable manifold at eriais is The total velocity for indirect cature is v. The total transfer time T includes the stable transfer time, arking time T and unstable transfer time T u, that is T Ts T Tu. In this aer, we focus on evaluate the efficiency of indirect cature. Therefore, the eriais states have not been considered excet the eriais distance. There are multile choices for arking orbits. However, numerical simulations show that different arking orbits cost nearly the same velocity so long as the eriais distances are identical. Three kinds of mission orbits are chosen (1) 200km circular orbit (2) ellitic orbit (3) 20000km T s v u. circular orbit. For each orbit, different excess velocity is investigated, 1.88 km / s, 2.09 km / s and v 3.39 km / s. Table 1-3 shows the detailed information. Table 1 Cost for different cature strategies (200km circular orbit) (km/s) Direct Cature Indirect cature (km/s) (km/s) (day) v d In first situation, the L2 lanar Lyaunov orbit with amlitude.7 10 km is chosen. The stable manifolds with eriais distance r 376km is chosen. The eriais velocity is v km / s. Stable transfer cost days. The unstable transfer time is about. Sacecraft will stay in arking orbit for 186.8days to insert into unstable manifold, the insert erturbation is only 1m/s. The velocity at unstable manifold s eriais is vu km / s. The transfer requires about days. As shown in Table.1, the cost for direct and indirect cature is similar at low altitude orbit. Though indirect cature cost long transfer time, it rovides a chance to exlore the sace environment in the vicinity of Mars and Lagrange oints without extra velocity increment. Table 2 Cost for different cature strategies (800*60000km orbit) Direct Cature Indirect cature vd v (km/s) (km/s) (km/s) (day) (km/s) v d In Table 2, we choose an ellitic orbit, which is more common in Mars exloration. The L2 Halo orbit with amlitude A km is selected. The eriais z distance for stable manifolds is r 394km, the total cature time is days includes days arking time. As the eriais of mission orbit increases, the indirect cature shows its advantages when comared with direct cature. Moreover, indirect cature saves more fuel when the excess velocity increases. v v Table 3 Cost for different cature strategies (20000km circular orbit) T T

8 Direct cature Indirect cature vd v (km/s) (km/s) (day) (km/s) (km/s) v d v In third situation, a high altitude orbit is investigated, which is similar to Mars geostationary orbit. Such orbits might be utilized for Mars navigation. The L1 halo orbit with amlitude Az km is used as arking orbit. It is clearly seen from Table 3 that indirect cature could save more than 30% velocity in such high orbit. The efficiency is even better in higher. Figure 13 shows the indirect cature trajectory in global and local view. T cature. There is no limitation to the cature altitude and excess velocity. Moreover, indirect cature shows better efficiency for high altitude and high orbit insertion. Besides the fuel saving, other advantages of indirect cature are discussed as follow, (1) Indirect cature takes full advantage of roerties of invariant manifolds. The sacecraft required no more maneuver during the manifolds transfer. That rovides a good oortunity for sace observation and environment exloration. (2) The arking orbit could achieve transfer to different mission orbits. Therefore, the sacecraft could choose mission orbit after detailed analysis during transfer and arking, which increases the flexibility of transfer. (3) Comared with direct cature, one large cature maneuver is relaced by two smaller cature maneuvers, which also reduce the gravity loss during cature. The indirect lanetary cature strategy is well suited for exloration mission which is insensitivity to time. It can also be alied to other lanets cature and rovide reference for future exloration mission. 6. CONCLUTION (a) In this aer, the indirect lanetary cature via eriodic orbit is investigated under the background of Mars exloration. The eriodic orbit is considered as a ark orbit during the cature, which connects with the interlanetary trajectory and mission orbit by stable and unstable manifolds, resectively. The orbit selection for lanar and satial eriodic orbit are investigated. The effect of indirect cature is evaluated under different scenarios. The result shows that indirect cature requires less velocity than direct cature. Better efficiency can be found for high altitude and high orbit cature. It also has the advantages in flexibility of transfer, extra scientific returns, which may be interested for future lanetary exloration missions. 7. ACKNOWLEDGEMENT This work was suorted by the Program for New Century Excellent Talents in University and the National Natural Science Foundation of China (Grant No ). 8. REFERENCES (b) Fig. 13 Indirect cature trajectory to 20000km circular orbit (a) global view (b) in the vicinity of the Mars Based on the discussion and evaluation, it is concluded that the indirect lanetary cature via eriodic orbit around Lagrange oints could save velocity increment than direct [1] Lyons, Daniel T., et al. "Mars global surveyor: aerobraking mission overview." Journal of Sacecraft and Rockets 36.3: , [2] Smith, John C., and Julia L. Bell. "2001 Mars Odyssey Aerobraking."Journal of sacecraft and rockets 42.3: , 200.

9 [3] Belbruno, Edward A., and James K. Miller. "Sun-erturbed Earth-to-Moon transfers with ballistic cature." Journal of Guidance, Control, and Dynamics16.4: , [4] Koon, Wang Sang, et al. "Low energy transfer to the Moon." Dynamics of Natural and Artificial Celestial Bodies. Sringer Netherlands, 63-73, [] Belbruno, Edward, Francesco Touto, and Marian Gidea. "Resonance transitions associated to weak cature in the restricted three-body roblem."advances in Sace Research 42.8: , [6] Chung, M. J., et al. "Trans-lunar cruise trajectory design of GRAIL (gravity recovery and interior laboratory) mission." Paer aiaa 8384, [7] Schoenmaekers, J., D. Horas, and J. A. Pulido. "SMART-1: with solar electric roulsion to the Moon." 16th International Symosium on Sace Flight Dynamics, Pasadena, California. Vol [8] Castillo, A., Belló, M., González, J. A., et al, Use of Weak Stability Boundary Trajectories for Planetary Cature, (2003). IAF, 4th International Astronautical Congress, Bremen, Germany, No. IAF-03-AP 31, Se. 29- Oct. 03, [9] Touto, Francesco, Massimiliano Vasile, and Franco Bernelli- Zazzera. "Low energy interlanetary transfers exloiting invariant manifolds of the restricted three-body roblem." Journal of the Astronautical Sciences 3.4: , 200. [10] Mingotti, Giorgio, Francesco Touto, and F. Bernelli- Zazzera. "Earth Mars transfers with ballistic escae and lowthrust cature." Celestial Mechanics and Dynamical Astronomy 110.2: , [11] Touto, Francesco, and Edward Belbruno. "Earth Mars transfers with ballistic cature." Celestial Mechanics and Dynamical Astronomy 121.4: , 201. [12] Broucke, R. "Traveling between the Lagrange oints and the Moon." Journal of Guidance, Control, and Dynamics 2.4: , [13] Lo, Martin W, and S. D. Ross. "Low Energy Interlanetary Transfers Using the Invariant Manifolds of L1, L2, and Halo Orbits." AAS/AIAA Sace Flight Mechanics Meeting, [14] Davis, Kathryn E., Rodney L. Anderson, and George H. Born. "Preliminary Study of Geosynchronous Orbit Transfers from LEO using Invariant Manifolds." The Journal of the Astronautical Sciences 8.3: , [1] Nakamiya, Masaki, et al. "Analysis of cature trajectories into eriodic orbits about libration oints." Journal of guidance, control, and dynamics 31.: , [16] Nakamiya, Masaki, et al. "Interlanetary transfers between halo orbits: connectivity between escae and cature trajectories." Journal of guidance, control, and dynamics 33.3: , [17] Wang, Yamin, Dong Qiao, and Pingyuan Cui. "Analysis of two-imulse cature trajectories into halo orbits of Sun Mars system." Journal of Guidance, Control, and Dynamics 37.3: ,