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Research ArticleNumerical Study of the Zero Velocity Surface and TransferTrajectory of a Circular Restricted Five-Body Problem
R. F. Wang and F. B. Gao
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
Correspondence should be addressed to F. B. Gao; gaofabao@sina.com
Received 14 May 2018; Revised 26 July 2018; Accepted 5 September 2018; Published 15 October 2018
Academic Editor: Chris Goodrich
Copyright Β© 2018 R. F. Wang and F. B. Gao. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
We focus on a type of circular restricted five-body problem in which four primaries with equal masses form a regular tetrahedronconfiguration and circulate uniformly around the center of mass of the system. The fifth particle, which can be regarded as asmall celestial body or probe, obeys the law of gravity determined by the four primaries. The geometric configuration of zero-velocity surfaces of the fifth particle in the three-dimensional space is numerically simulated and addressed. Furthermore, a transfertrajectory of the fifth particle skimming over four primaries then is designed.
1. Introduction
The restricted N-body problem has attracted the attention ofmany researchers from the fields of mathematics, astronomy,and mechanics because of its wide application in deepspace exploration. Here we list some recent or interestingresearch results. Gao and Zhang [1] studied the existence ofperiodic orbits of the circular restricted three-body problem.According to the existing literature, the first type of Poincareperiodic orbit generally requires that the mass parameter πof the system is sufficiently small, and the periodic orbitstudied in this paper is applied to any π between (0, 1), solvingthe problem that the first type of Poincareβs periodic orbithas always been considered to occur only when the massesof primaries are quite different. Baltagiannis and Papadakis[2] obtained the zero-velocity surfaces and correspondingequipotential curves in the planar restricted four-body prob-lem where the primaries were always at the vertices of anequilateral triangle. Alvarez-RamΔ±rez and Vidal [3] analyzedthe zero-velocity surfaces and zero-velocity curves of thespatial equilateral restricted four-body problem. Singh [4]investigated the permissible regions of motion and the zero-velocity surfaces under the influence of small perturbations inthe Coriolis and centrifugal forces in the restricted four-bodyproblem.Mittal et al. [5] found the zero-velocity surfaces andregions of motion in the restricted four-body problem with
variablemasswhere the three primaries formed an equilateraltriangle. Asique et al. [6] drew the zero-velocity surfaces todetermine the possible permissible boundary regions in thephotogravitational restricted four-body problem.
However, limited research studies have been performedon circular restricted five-body problem while it is comparedwith related three-body and four-body problems. A spatialcircular restricted five-body problem wherein the fifth parti-cle (the small celestial body or probe) with negligible mass ismoving under the gravity of the four primaries, which movein circular periodic orbits around their centers of mass fixedat the origin of the coordinate system. Because the mass ofthe fifth particle is small, it does not affect the motion of fourprimaries.
Kulesza et al. [7] observed the region of motion of therestricted rhomboidal five-body problem whose configura-tion is a rhombus using theHamiltonian structure and provedthe existence of periodic solutions. Albouy and Kaloshin [8]confirmed there were a finite number of isometry classes ofplanar central configurations, also called relative equilibria,in the Newtonian five-body problem. Marchesin and Vidal[9] determined the regions of possible motion in the spa-tial restricted rhomboidal five-body problem by using theHamiltonian structure. Llibre and Valls [10] found that theunique cocircular central configuration is the regular 5-gon
HindawiMathematical Problems in EngineeringVolume 2018, Article ID 7489120, 10 pageshttps://doi.org/10.1155/2018/7489120
2 Mathematical Problems in Engineering
with equal masses for the five-body problem. Bengochea etal. [11] studied the necessary and sufficient conditions forperiodicity of some doubly symmetric orbits in the planar1 + 2π-body problem and studied numerically these typesof orbits for the case n=2. Shoaib et al. [12] considered thecentral configuration of different types of symmetric five-body problems that have two pairs of equal masses; thefifth mass can be both inside the trapezoid and outside thetrapezoid, but the triangular configuration is impossible. Xuet al. [13] discussed the prohibited areas of the Sun-Jupiter-Trojans-Greeks-Spacecraft system and designed a transfertrajectory from Jupiter to Trojans. Han et al. [14] obtainedmany new periodic orbits in the planar equal-mass five-bodyproblem by using the variational method. Gao et al. [15]investigated the zero-velocity surfaces and regions of motionfor specific configurations in the axisymmetric restrictedfive-body problem. Saari and Xia [16] verified that, withoutcollisions, the Newtonian N-body problem of point massescould eject a particle to infinity in finite time and that three-dimensional examples exist for all π β©Ύ 5.
In the present paper, it is assumed that the four primarieswith equal masses constitute a regular tetrahedron configura-tion. A dynamic equation of the circular restricted five-bodyproblem is established, and the relationship between energysurface structure of the fifth particle and the correspondingJacobi constant is discussed. Moreover, the critical positionof the fifth particleβs permissible and forbidden regions ofmotion is also addressed. In addition, based on Matlabsoftware, a transfer trajectory of the fifth particle skimmingover four primaries is designed numerically. Because ofthe gravity of the four primaries, the transfer trajectorywill reduce the consumption of fuel for the fifth particleeffectively.
2. Equations of Motion
For a circular restricted five-body system, assume each ofthe four primaries, namely, π1, π2, π3, and π4, lie at oneof the vertices of a regular tetrahedron. Furthermore, theirmotions are opposite to the center of mass. The small orbiterπ is only subjected to the gravity of the four primaries,and the origin π of the inertial coordinate system πππ islocated at the center of mass of the four primaries, with oneof them, say π1, located on the π-axis. The plane defined bythe remaining three primaries is parallel to πππ plane. Theorigin π of the rotational coordinate system π₯π¦π§ coincideswith π. The π₯-axis and π¦-axis of the rotational coordinatesystem are rotated counterclockwise around the centroid ofthe unit angular velocity relative to the π-axis and π-axis ofthe inertial coordinate system πππ, respectively.
Suppose the masses of four primaries are 1/4, the massof the fifth particle is π, the angle that the system rotatesaround its center of mass is π, and the two coordinatesystems coincide with each other when the time π‘ is 0.Hence, we obtain π is equal to π‘. The dimensionless distancebetween each two primaries is 1, and the distances betweenfour primaries and the center of mass are β6/4. Thus, thecoordinates of four primaries are
π1 (0, 0, β64 ) ,π2 (ββ33 cos (π6 β π‘) , β33 sin (π6 β π‘) , ββ612 ) ,π3 (β33 cos(π6 + π‘) , β33 sin (π6 + π‘) , ββ612 ) ,
π4 (β33 cos (2π3 β π‘) , ββ33 sin (2π3 β π‘) , ββ612 ) .
(1)
In the coordinate system π₯π¦π§, we hypothesize the coordinateof π is (π₯, π¦, π§); thus, the coordinates of four primaries are
π1 (0, 0, β64 ) ,π2 (β12 , β36 , ββ612 ) ,π3 (12, β36 , ββ612 ) ,π4 (0, ββ33 , ββ612 ) .
(2)
Suppose that the coordinate of the fifth particle is(π, π, π) in the inertial coordinate system; thus, the Lagrangefunction satisfies the following relationship:
πΏ = 12 (οΏ½οΏ½2 + οΏ½οΏ½2 + οΏ½οΏ½2) β π (π, π, π, π‘) , (3)
where the gravitational potential energyπ(π, π, π) is definedas
π (π, π, π, π‘) = β14 ( 1π 1 + 1π 2 + 1π 3 + 1π 4) , (4)
and π π(π = 1, 2, 3, 4) denotes the distance between the fifthparticle and one primary:
π 1 = βπ2 + π2 + (π β β64 )2,π 2 = β(π + β33 cos (π6 β π‘))2 + (π β β33 sin (π6 β π‘))2 + (π + β612 )2,
Mathematical Problems in Engineering 3
π 3 = β(π β β33 cos (π6 + π‘))2 + (π β β33 sin (π6 + π‘))2 + (π + β612 )2,π 4 = β(π β β33 cos (2π3 β π‘))2 + (π + β33 sin (2π3 β π‘))2 + (π + β612 )2.
(5)
By substituting (3) into the following Euler-Lagrange equa-tion,
πππ‘ β ππΏπππ β ππΏπππ = 0, (6)
we obtain the dimensionless equations of motion of the fifthparticle in the inertial coordinate system
οΏ½οΏ½ = βππ,οΏ½οΏ½ = βππ,οΏ½οΏ½ = βππ,(7)
whereππ,ππ, andππ denote the derivative ofπwith respectto π, π, and π, respectively.
Suppose that the coordinates of the orbiter π in therotating coordinate system are (π₯, π¦, π§). The relationshipbetween the two coordinate systems is
(πππ) = π π‘(π₯π¦π§) , (8)
where π π‘ isπ π‘ = (cos π‘ β sin π‘ 0
sin π‘ cos π‘ 00 0 1) . (9)
Thus, the dimensionless equations of motion of the fifthparticle in the rotational coordinate system are
π₯ β 2οΏ½οΏ½ = Ξ©π₯,π¦ + 2οΏ½οΏ½ = Ξ©π¦,οΏ½οΏ½ = Ξ©π§,(10)
where the generalized potential energy Ξ©(π₯, π¦, π§) is definedas
Ξ© (π₯, π¦, π§) = 12 (π₯2 + π¦2) + 14 ( 1π1 + 1π2 + 1π3 + 1π4) , (11)
and
π1 = βπ₯2 + π¦2 + (π§ β β64 )2,π2 = β(π₯ + 12)2 + (π¦ β β36 )2 + (π§ + β612 )2,π3 = β(π₯ β 12)2 + (π¦ β β36 )2 + (π§ + β612 )2,π4 = βπ₯2 + (π¦ + β33 )2 + (π§ + β612 )2.
(12)
Thus, a first Jacobi type integral is
2Ξ© (π₯, π¦, π§) β (οΏ½οΏ½2 + οΏ½οΏ½2 + οΏ½οΏ½2) = πΆ, (13)
where π = βοΏ½οΏ½2 + π¦2 + οΏ½οΏ½2 is the motion velocity of the fifthparticle and πΆ is the Jacobi constant. The permissible motionregion and prohibited region are defined byπ β©Ύ 0 and π < 0,respectively. Therefore, when the velocity of the fifth particleis zero, the curve shown by the above equation is called zero-velocity curve on the plane and is called zero-velocity surfacein space.
3. Zero-Velocity Surfaces
The diagrams of relationship between the zero-velocity sur-faces and the Jacobi constant πΆ are shown in the following.
For a given value of πΆ, we can obtain the zero-velocitysurfaces and the zero-velocity curves of the circular restrictedfive-body problem on the π₯ππ¦, π₯ππ§, and π¦ππ§ planes, as shownin Figures 1(a)β1(c), respectively.
We now turn to discuss the zero-velocity surfaces of thefifth particle.
For the Jacobi type integral (13) of the system, when thevelocity of the fifth particle is zero, the relationship betweenthe zero-velocity surface and the values of Jacobi constant πΆis shown in Figures 1(a)β1(c). The smaller the πΆ value, thegreater the energy of the system. In addition, the area of thezero-velocity surface of the fifth particle decreases, while thepermissible motion region of the fifth particle increases.
Figure 2 shows the evolution of prohibited area of the fifthparticle when πΆ is 3.4. The fifth particle can only skim over
4 Mathematical Problems in Engineering
3
2.5
2
1.5
1
0 0
2
2
y xβ2 β2
C
(a)
321
0
0
0
x
β2.5
4
5
5
3
2
1
2.5
z β5β3 β2 β1
C
(b)
032
10
yβ2
5
z β5 β3β1
0
4
5
3
2
1
C
(c)
Figure 1: (a), (b), and (c) are the diagrams of the relationships between the zero-velocity surfaces of the circular restricted five-body problemand the Jacobi constant πΆ on the π₯ππ¦, π₯ππ§, and π¦ππ§ planes, respectively.
0
00
xy
4
2
2
2
1
1
β2
β2 β2
β4
β1
β1
z
Figure 2: Zero-velocity surface of the fifth particle: πΆ = 3.4.four primaries under the gravity, rather than shuttle amongthem.
When πΆ is 3.4, as shown in Figure 3, the regions that thefifth particle can move around is the neighborhood of thosethree small circles on π₯ππ¦ and π₯ππ§ planes under the gravity offour primaries, while it moves around those two small circleson the π¦ππ§ plane.
When πΆ is 3.2829, as shown in Figures 4(a) and 4(b),the permissible regions of the two primaries, π2 and π3,are interconnected to create βchannelβ. Therefore, βchannelAβ appears, through which fifth particle can pass through.However, it cannot fly across the permissible area of π1. Theprohibited areas of the fifth particle decrease.
WhenπΆ is 3.2828, as shown in Figures 5(a) and 5(b), thereare two new channels, namely, βchannel Bβ and βchannel Cβ,through which the fifth particle can fly from the permissiblearea of π4 into π2 and π3, respectively. The prohibitedregions of the fifth particle decrease.
The prohibited area of the fifth particle when πΆ reducesto 3.2774 is presented in Figures 6(a) and 6(b). βChannel Dβand βchannel Eβ are the channels through which the fifthparticle can fly from the permissible area of π1 into π2, π3,respectively.
When πΆ is 3.2772, there is a βchannel Fβ, through whichthe fifth particle can fly from the permissible area of π1 intoπ3, as shown in Figures 7(a) and 7(b). At this point, the fifthparticle can travel through the above six channels between thepermissible regions of each two interconnected primaries.
When πΆ is 3.2675, with the larger βchannel Dβ andβchannel Eβ, a new βchannel Gβ is formed, as shown in Figures8(a) and 8(b). Moreover, if πΆ is further reduced, the fifth
Mathematical Problems in Engineering 5
x
y
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
xz
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
y
z
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
Figure 3: Zero-velocity curves of the fifth particle for πΆ = 3.4.
00
0
x
2
2
2
1
1
β2
β2
β3
3
β2β1
β1
z
y
(a) (b)
Figure 4: (a). Zero-velocity surface of the fifth particle: πΆ=3.2829. (b) Magnified view of the zero-velocity surface: πΆ=3.2829.particle will move from the permissible region of π2 to π3directly, without any help of other channels.
As shown in Figures 9(a) and 9(b), βchannel Hβ andβchannel Iβ exist when C is 3.2674. Thus, with πΆ furtherdeclining, the fifth particle will move from the permissiblearea of π4 to π2 or π3 through π1; i.e., its prohibited areasare reduced.
When πΆ is 3.2674, as shown in Figure 10, the fifth particlecan move from the inside of these βpetalsβ, except the centerof the outer ring, to periphery of the βpetalsβ onπ₯ππ¦ plane andmove from the inside of the βcloverβ to the periphery of theβcloverβ directly on the π₯ππ§ plane. On the π¦ππ§ plane, there is apoint of intersection, that is, βfortress Tβ, where fifth particlecan fly through it. Moreover, the prohibited areas of the fifthparticle decrease with the decrease of πΆ value.
Figures 11(a) and 11(b) show the prohibited area of thefifth particle when πΆ is 3.2662. The βregion Jβ will disappearwith the decrease of πΆ. At this case, the fifth particle will bein the permissible region of four primaries within the shuttleflight, without any assistance of the βchannelsβ. However, itstill cannot fly into outer space.
As shown in Figures 12(a) and 12(b), when πΆ is 3.2085,the permissible regions of π1, π2, and π3 communicate thefeasible regions of the system, growing three new βchannelsβ,
namely, K, N, and L, through which the fifth particle can flyto outer space.
WhenπΆ is 2.8383, the prohibited areas of the fifth particleare further reduced in Figures 13(a) and 13(b). With πΆ valuefurther decreasing in Figure 13(b), the prohibited areas P, Q,and R will disappear, and the fifth particle will not need tocrossover any βchannelsβ to fly into outer space. Thus, theprohibited areas will gradually shrink to a point and finallydisappear. The fifth particle will be free of the influences ofthe four primaries and be able to fly into outer space.
As shown in Figure 14, the prohibited areas of the fifthparticle appear as three circles on the π₯ππ¦ plane, and theprohibited areas on the π₯ππ§ plane are two parts of upper andlower concave and convex curves. In addition, a new βfortressWβ, through which the fifth particle can fly to outer space,appears on the π¦ππ§ plane when πΆ is 2.8383. As C decreases,the prohibited areas on these three planes will disappear.Finally, the fifth particle will not be bound by the primariesand will be able to fly freely on these planes.
4. Numerical Simulation ofthe Transfer Trajectory
In deep space exploration, Figure 9 plays a role in the circularrestricted five-body problem, because when Jacobi constant
6 Mathematical Problems in Engineering
00
0
x
2
2
2
1
1
β2
β2
β3
3
β2 β1
β1
z
y
(a) (b)
Figure 5: (a). Zero-velocity surface of the fifth particle: πΆ=3.2828. (b) Magnified view of the zero-velocity surface: πΆ=3.2828.
0
0
0
x
2
2
2
1
1
β2
β2
β3
3
β2 β1
β1
z
y
(a) (b)
Figure 6: (a) Zero-velocity surface of the fifth particle: πΆ = 3.2774. (b) Magnified view of the zero-velocity surface: πΆ = 3.2774.
0
0
0
x
2
2
2
1
1
β2
β2
β3
3
β2 β1
β1
z
y
(a) (b)
Figure 7: (a). Zero-velocity surface of the fifth particle: πΆ = 3.2772. (b) Magnified view of the zero-velocity surface: πΆ = 3.2772.
Mathematical Problems in Engineering 7
0
0
0
x
2
2
1
1
β2
β5
5
β2 β1
β1
z
y
(a) (b)
Figure 8: (a) Zero-velocity surface of the fifth particle: πΆ = 3.2675. (b). Magnified view of the zero-velocity surface: πΆ = 3.2675.
0
0
0
x
2
2
1
1
β2
β5
5
β2β1
β1
z
y
(a) (b)
Figure 9: (a) Zero-velocity surface of the fifth particle: πΆ = 3.2674. (b) Magnified view of the zero-velocity surface: πΆ = 3.2674.
x
y
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
x
z
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
y
z
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
T
Figure 10: Zero-velocity curves of the fifth particle for πΆ = 3.2674.value C is greater than that in Figure 9, the fifth particlemust fly from the permissible area of one primary to anotherthrough the corresponding βfortressesβ. With the decreaseof the πΆ value, the permissible areas of the fifth particlewill increase, allowing the fifth particle to be able to shuttlebetween the permissible regions of the four primaries.
The calculation shows that the four primariesπ1, π2, π3, and π4 locate at (0, 0, 0.61237),(β0.5, 0.288675, β0.204124), (0.5, 0.288675, β0.204124), and(0, β0.57735, β0.204124), respectively. With appropriateinitial conditions used, the numerical method is adopted
to simulate the motions of the circular restricted five-bodyproblem based on Matlab 2012a and the algorithm of ode45which gives the fifth-order accurate method. Based on thesimulations, the following transfer trajectory is designed.
As shown in Figure 15, the fifth particle starts from thepoint A (i.e., π2) in the permissible region of primary π2,passes through B, C (two interior points of the primaryπ3), D, E (two interior points of the primary π4), and F(an interior point of the primary π1), and finally flies to G(a nearby point of π1). The specific analysis is as follows:first, for the primary π2 (i.e., the point A), we consider
8 Mathematical Problems in Engineering
0
0
0
x
2
2
1
1
β2
β5
5
β2
β1
β1
z
y
(a) (b)
Figure 11: (a) Zero-velocity surface of the fifth particle: πΆ = 3.2662. (b) Magnified view of the zero-velocity surface: πΆ = 3.2662.
0
0
0
x
2
2
2
1
1
β2
β2
β3
3
β2β1
β1
z
y
(a) (b)
Figure 12: (a) Zero-velocity surface of the fifth particle: πΆ = 3.2085. (b) Magnified view of the zero-velocity surface: πΆ = 3.2085.
02 0
0
x2
2
1
1
β2
β2
β3
3
β1
β1
z
y
(a) (b)
Figure 13: (a) Zero-velocity surface of the fifth particle: πΆ = 2.8383. (b) Magnified view of the zero-velocity surface: πΆ = 2.8383.the initial conditions π₯0 = [β0.5, 0.288675, β0.204124],V0 = [0.76, 0.009, β0.02], where π₯0 is the initial posi-tion coordinates of the fifth particle, and V0 is its ini-tial velocity. By numerical simulation, we obtain a trans-fer trajectory, on which the fifth particle can fly fromπ2(β0.5, 0.288675, β0.204124), starting at the initial speed
(0.76, 0.009, β0.02), to finally reach the permissible region ofπ3, i.e., a track from point A to point B. Second, selectingπ₯0 = [0.4705, 0.07041, 0.04673], V0 = [0.06, 0.09, 0.02],a permissible area of the internal transfer trajectory in theprimary π3 is the fifth particles track from B to C. Atthis case, C is very close to π3. Third, choosing π₯0 =
Mathematical Problems in Engineering 9
x
y
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
xz
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
y
z
β2 β1 0 1 2β2
β1.5
β1
β0.5
0
0.5
1
1.5
2
W
Figure 14: Zero-velocity curves of the fifth particle for πΆ = 2.8383.
0
0
0
x
2
2
2
1
1
1
β2
β2
β3
3
β2 β1
β1
β1
z
y
(a) (b)
Figure 15: (a) The orbits of motion of the fifth particle skimming over four primaries. (b) Magnified view of these orbits.
[0.5233, 0.2849, β0.1971], V0 = [β1.3, β5.7, 1.9] correspondsto a transfer trajectory from C to D; at the same time,the fifth particle flies from the permissible regions of π3to π4. Fourth, for π₯0 = [β0.2277, β0.4345, 0.008506],V0 = [0.2, 0.07, 0.008], we obtain a transfer trajectory fromD to E inside the active area of the primary π4, withthe point E and π4 being very close. Fifth, for π₯0 =[0.002021, β0.5811, β0.2055], V0 = [7.3, 5.9, 9.8], the fifthparticle can fly from E to F at the flying track; i.e., thefifth particle moves from the permissible region of π4to π1. Sixth, with π₯0 = [0.1188, 0.1583, 0.3905], V0 =[0.01, 0.001, 0.3], we obtain an orbit where the fifth particleflies from F to G inside the permissible area of primary π1,and point G is very close to π1. Finally, a complete transfertrajectory is obtained, when the flyby occurs between the fifthparticle and the four primaries.
5. Conclusions
In this paper, we introduced and established the dynamicequations of the circular restricted five-body problem, andthe geometric configurations of zero-velocity surfaces andzero-velocity curves of the problem with different Jacobiconstants of πΆ values are also discussed in detail. Through
the analysis and discussion of zero-velocity surfaces and zero-velocity curves of the system, we found that, when the πΆvalue is large, the performance of the fifth particle is not veryactive. When πΆ decreased, the prohibited areas of the fifthparticle decreased, while the permissible regions expanded,ultimately eliminating the possibility of the four primariesflying freely in outer space. Based on these numerical results,the case of Figure 9 is found to be worthy of further study,because the fifth particle shuttled in the permissible regionswithin the primaries under the gravity, without too muchextra energy.Thefifthparticle can skimover to four primariesto explore them. Therefore, we simulated the correspondingtransfer trajectories numerically, starting from the primaryπ2, followed by π3, π4, and finally to fly around π1,to explore the transfer trajectories of four primaries; thisapproach will not only improve the exploration efficiency butalso save resources.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
10 Mathematical Problems in Engineering
Acknowledgments
The authors gratefully acknowledge the support of theNational Natural Science Foundation of China (NSFC)through Grants nos. 11672259, 11302187, and 11571301; theMinistry of Land and Resources Research of China in thePublic Interest through Grant no. 201411007; and the TopNotch Academic Programs Project of Jiangsu Higher Educa-tion Institutions (TAPP) through Grant no. PPZY2015B109.
Supplementary Materials
There are three MATLAB code files in the supplemen-tary material, namely, βeqns.mβ, βfifth-body.mβ, and βtrajec-tory.mβ. The file βEqn.mβ contains the governing equation ofthe fifthparticle, the file βfifth-body.mβ includes an extrapola-tionmethod for this problem, and themain file βtrajectory.mβis used to design the transfer trajectories between the fourprimaries. (Supplementary Materials)
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