on the construction of low-energy cislunar and translunar transfers based on the libration points

Astrophys Space Sci (2013) 348:65–88DOI 10.1007/s10509-013-1563-6

O R I G I NA L A RT I C L E

On the construction of low-energy cislunar and translunartransfers based on the libration points

Ming Xu · Yan Wei · Shijie Xu

Received: 4 March 2013 / Accepted: 8 July 2013 / Published online: 9 August 2013© Springer Science+Business Media Dordrecht 2013

Abstract There exist cislunar and translunar librationpoints near the Moon, which are referred to as the LL1

and LL2 points, respectively. They can generate the differ-ent types of low-energy trajectories transferring from Earthto Moon. The time-dependent analytic model includingthe gravitational forces from the Sun, Earth, and Moon isemployed to investigate the energy-minimal and practicaltransfer trajectories. However, different from the circularrestricted three-body problem, the equivalent gravitationalequilibria are defined according to the geometry of theinstantaneous Hill boundary due to the gravitational per-turbation from the Sun. The relationship between the alti-tudes of periapsis and eccentricities is achieved from thePoincaré mapping for all the captured lunar trajectories,which presents the statistical feature of the fuel cost andcaptured orbital elements rather than generating a specifiedMoon-captured segment. The minimum energy required bythe captured trajectory on a lunar circular orbit is deducedin the spatial bi-circular model. The idea is presented thatthe asymptotical behaviors of invariant manifolds approach-ing to/traveling from the libration points or halo orbits aredestroyed by the solar perturbation. In fact, the energy-minimal cislunar transfer trajectory is acquired by transitingthe LL1 point, while the energy-minimal translunar transfertrajectory is obtained by transiting the LL2 point. Finally,the transfer opportunities for the practical trajectories thathave escaped from the Earth and have been captured by theMoon are yielded by the transiting halo orbits near the LL1

and LL2 points, which can be used to generate the whole ofthe trajectories.

M. Xu (B) · Y. Wei · S. XuDepartment of Aerospace Engineering, School of Astronautics,Beihang University, Beijing 100191, Chinae-mail: [email protected]

Keywords Libration point · Halo orbit · Hamiltoniansystem · Low-energy cislunar trajectory · Low-energytranslunar trajectory · Weak-stability-boundary transfer

1 Introduction

Previous research on cislunar transfer trajectories fromthe Earth to Moon in the context of two-body dynam-ics reached the conclusion that the spacecraft has to beaccelerated up to the hyperbolic velocity so as to escape theEarth’s gravitational force; while some recent research fromthe viewpoint of the restricted circular three-body prob-lem (abbreviated CR3BP) showed that the hyperbolic ve-locity is not a necessary condition for the cislunar trans-fer (Koon et al. 2007). Compared with Hohmann transfer,the ballistically captured trajectory known as the type oflow-energy transfer trajectories (Xu and Xu 2009), whichis obtained within the context of CR3BP, has lower fuelconsumption but longer transfer duration.

Conley studied the local dynamical behavior of planarCR3BP near the collinear libration point and classified allthe trajectories into four different types as follows: periodi-cal orbit (named Lyapunov orbit), stable/unstable manifoldsof periodic orbit, transiting, and non-transiting trajectories(Conley 1968). It is concluded from Conley’s work that theinvariant manifolds of periodic orbits will separate transit-ing and non-transiting trajectories, and only the transitingones can be employed to generate the low-energy cislunartransfer trajectories.

McGehee investigated the global dynamical behaviorof CR3BP and achieved similar results, i.e., the stableand unstable manifolds of Lyapunov orbit form a two-dimensional hyper-surface in the three-dimensional Eu-clidean space which may play a significant role in under-standing the transiting trajectories (McGehee 1969). Based

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66 Astrophys Space Sci (2013) 348:65–88

on the preliminary work of Conley and McGehee, Marsdenand Ross extended and thoroughly investigated the dynami-cal structure near the libration point, and denoted the invari-ant manifolds as Conley–McGehee tubes (abbreviated C-Mtubes) in order to acknowledge the contributions of these au-thors (Marsden and Ross 2006). Yamato demonstrated thatmost of the tubes are distorted but few of them are preservedby small perturbations from the perturbed gravitation of thethird celestial body (Yamato 2003).

Several scholars devoted their work to the topic of sometransiting trajectories near the LL1 point, since Conley hadachieved the low-energy cislunar trajectories from the view-point of the LL1 point (Conley 1969). Bolt and Meissobtained a low-energy cislunar transfer trajectory by theshooting method developed in chaotic dynamics with a totalfuel consumption of �V = 750 m/s and a flight durationof �t = 748 days (Bollt and Meiss 1995). Schroer and Ottimproved the shooting method to achieve the transfer tra-jectory with similar fuel consumption but cutting off half ofthe transfer time (�t = 377.5 days) (Schroer and Ott 1997).Macau gained a transfer trajectory with a little more fuelconsumption but much less transfer time than Schroer andOtt, i.e., �V = 767 m/s and �t = 284 days (Macau 1998).Ross and Koon optimized the transfer time and fuel con-sumption to yield a better result, i.e., �V = 860 m/s and�t = 65 days (Ross and Koon 2003). Topputo and Vasileemployed the Lambert equation in CR3BP to solve the two-point boundary problems and obtained a similar result asRoss and Koon (Topputo and Vasile 2005). Xu et al. investi-gated the condition of the occurrence of low-energy transferand discovered that the transiting trajectories near the LL1

point are preferred to generate a low-thrust cislunar trajec-tory (Xu et al. 2012).

On the other hand, Belbruno et al. introduced a newtype of translunar trajectories by numerical methods, whichfound a great application in rescuing Japanese lunar space-craft “Hiten” in 1991, and is referred to as the weak-stability boundary (abbreviated WSB) trajectory (Belbrunoand Miller 1993; Belbruno 2004). The WSB trajectory isconsidered as a significant contribution to celestial mechan-ics, and more analytic or semi-analytic investigations wereimplemented in this theory by Circi and Teofilatto (2001),Yagasaki (2004), Parker and Lo (2005), and García andGómez (2007).

Koon et al. investigated the long-term evolutions of C-Mtubes under the gravitational perturbation from the Sun,and divided the restricted four-body problem into two dif-ferent CR3BPs, i.e., the Sun-Earth/Moon system and theEarth–Moon system (Koon et al. 2001). A magic result wasachieved: that a Belbruno WSB trajectory can be generatedfrom the stable manifolds near the EL1 (or EL2) point andthe unstable manifolds near the translunar LL2 point, withthe assistance of the numerical tool of Poincaré mapping.

Different from the above research, focusing only on spec-ified Earth-to-Moon transfer trajectories, a systematic dis-cussion of both cislunar and translunar trajectories is pre-sented in the context of restricted four-body dynamics inthis paper. The statistical features of the fuel cost and cap-tured orbital elements, like altitude of periapsis and ec-centricity, are investigated by the tool of Poincaré map-ping rather than a specified Moon-captured segment. Com-pared to CR3BP and the Hill model, both the cislunar andtranslunar trajectories with the minimum energy are de-duced in a spatial analytical four-body model including thegravitational forces from the Sun, Earth, and Moon. Westudy how the asymptotical behaviors of invariant mani-folds approaching to/traveling from libration points or haloorbits are destroyed in the time-independent model. Theenergy-minimal and practical cislunar transfer trajectoriesare acquired by transiting the LL1 point and halo orbitsnear the point, respectively; however, the energy-minimaland practical translunar transfers are obtained by transitingthe LL2 point and halo orbits near the point. Furthermore,the transfer opportunities for the practical trajectories thathave escaped from the Earth and have been captured by theMoon are yielded by the transiting halo orbits near the LL1

and LL2 points. This can be used to generate the whole ofthe trajectories.

2 Lunar captured trajectories in spatial bi-circularmodel

Compared with the Hohmann direct transfer employed bythe Apollo (NASA) and Chang’E (China) missions, thelow-energy WSB transfer requires more fuel in the accel-erating maneuvers, and then much less fuel cost in thedecelerating maneuvers, which will make the WSB type oflunar transfer trajectories more economic than the Hohmanntype trajectories. Therefore, Belbruno and Miller (1993),and García and Gómez (2007) proposed the concept of lu-nar temporary captured trajectories to measure the opportu-nity of a spacecraft for transferring from the Earth to Moon,which has a somewhat higher energy than the libration pointLL1 or LL2. When the spacecraft on the Hohmann trajectoryarrives at the Moon, its flight velocity is hyperbolical and itsosculating eccentricity is greater than 1, hence the spacecraftowns has higher energy than the LL2 point. However, thelow-energy trajectories are elliptical since their osculatingeccentricities are less than 1 during the flight, so that thespacecraft will keep orbiting the Earth with several loopsbefore transiting the libation point, and also keep orbitingthe Moon after transiting the point. Thus, the fuel cost of thelunar temporary capture turning into permanent capture issmaller than the Hohmann transfer.

An analytic spatial bi-circular model (abbreviated SBCM)including the gravitational forces from the Sun, Earth, and

Astrophys Space Sci (2013) 348:65–88 67

Fig. 1 The geometrical view of the SBCM model: the inclination ofthe lunar plane relative to the ecliptic plane is considered with an aver-age angle of 5◦9′; the solar phasic angle β measures the angle betweenthe line from the Earth to Moon and the intersecting line of the eclipticand lunar planes. The lunar phasic angle θs measures the angle betweenthe line from the Sun to the barycenter of the Earth–Moon system andthe intersecting line of the ecliptic and lunar planes. The ecliptic planeis painted in yellow, while the lunar plane is painted in green

Moon is developed in this section, and then a systemati-cal discussion of Moon-captured energy in this model isimplemented by the tool of numerical Poincaré mapping;however, no specific trajectory is referred to in this section.

2.1 The definition of SBCM

The SBCM originates from the planar bi-circular modeldeveloped by Koon et al. (2001) and the quasi-bi-circularmodel by Andreu (1999); specially, the SBCM showssignificant improvements in the inclination between theecliptic and lunar planes. Compared with the three modelsreferred to above, the SBCM has the following assumptions:(i) The Earth and Moon act as different simple gravitationalpoints, and move around their barycenter in Kepler circularmotions with their eccentricities ignored; (ii) the barycenterof the Earth–Moon system stays circumsolar in the eclipticplane with its eccentricity ignored; (iii) the inclination of thelunar plane relative to the ecliptic plane is considered withan average angle of 5◦9′.

In order to reduce the computational work in Kepler cir-cular motions under the Sun–Earth/Moon and Earth–Moonsystems, three different coordinates are introduced in thispaper, as shown in Fig. 1. The inertial IS-E/M frame withits components (X,Y,Z) is defined as follows: the originis fixed at the barycenter of the Sun–Earth/Moon system,and the X axis is along the intersection of the ecliptic andlunar planes, which follows an inertial direction in the sys-tem, and the Z axis is perpendicular to the lunar plane andalong the revolution axis of the Earth–Moon system, and theY axis is determined by the right-hand-side rule. Inheritingfrom IS-E/M, for a new inertial frame IE-M one has the samedefinition of the three axes, but one fixes its origin at thebarycenter of the Earth–Moon system. The syzygy SS-E/M

frame with its components (ξ, η, ζ ) is defined as follows:

the origin is fixed at the barycenter of the Sun–Earth/Moonsystem, and the ξ axis points from the Sun to the barycenterof the Earth and Moon. The ζ axis is perpendicular to theecliptic orbital plane, and the η axis is determined by theright-hand-side rule. The syzygy SE-M frame with its com-ponents (x, y, z) is defined as follows: the origin is fixed atthe barycenter of the Earth–Moon system, and the x axispoints from the Earth to the Moon. The z axis is perpen-dicular to the lunar plane and along the revolution axis ofthe Earth–Moon system, and the y axis is determined by theright-hand-side rule.

The equations derived in this paper can be normalizedby means of the characteristic length, time, and mass, asfollows:⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

[L] = LE-M, average distance between the Earthand Moon,

[M] = mE + mM, total mass in theEarth–Moon system,

[T ] = [R3

E-M/G(mE + mM)]1/2

,

where mE and mM are the mass of the Earth and Moon,respectively; LE-M is the average distance between the Earthand Moon; G is the universal gravitation constant.

RI = [X Y Z]T , � = [ξ η ζ ]T and r = [x y z]T are de-fined as the position vector of the spacecraft in the rotat-ing frames IS-E/M, SS-E/M, and SE-M, respectively. There-fore, the position vector from the Sun to the origin ofthe barycenter in the frame IS-E/M can be expressed asAS = (1 − μS)aS[cos θS sin θS 0]T , where aS (= 388.81114in the length unit normalization mentioned above) is theaverage distance between the heliocenter and the barycenterof the Earth–Moon system, and μS (= 3.040357143×10−6)

is the mass ratio of the Earth–Moon system with respect tothe full Sun–Earth–Moon system. i (= 5◦9′) is the inclina-tion between the ecliptic and lunar planes. θs is defined asthe lunar phasic angle measured between the line from theSun to the barycenter of the Earth–Moon system and theintersecting line of the ecliptic and lunar planes, and β isdefined as the solar phasic angle measured between the linefrom the Earth to Moon and the intersecting line. In this pa-per, the initial lunar angle θs0 is set as 0◦ at the momentt0 = 0, while the initial value of the solar angle β0 is selectedas the time variable to investigate Earth-to-Moon transfers inthe time-dependent SBCM model in the following sections.

According to the defined coordinate systems and SBCMassumptions, the required relationship between the space-craft’s position vector r, �, and RI is listed as

� = Rz(θS)RI, (1)

RI = Rx(−i)Rz(−β)r + AS, (2)

where Rz(θ) and Rx(θ) are the elementary transformationmatrices around the Z (or z) and X (or x) axes, respectively.


Thus, the Newtonian dynamics in the unit normalization isformulated as follows:

RI = −(1 − μ) · r − rE

‖r − rE‖3− μ · r − rM

‖r − rM‖3

− mS� − �S

‖� − �S‖3, (3)

where μ (= 0.0121516) is the mass ratio of the Moonwith respect to the Earth–Moon system, and �S = aS ·[−μS 0 0]T is the position vector of the Sun in the frameSS-E/M. The kinematics formulated by Eqs. (1) and (2) canbe used to deduce the dynamical equation:

⎡

⎣x

y

z

⎤

⎦ = −2

⎡

⎣−y

x

0

⎤

⎦ +⎡

⎣x

y

0

⎤

⎦ − (1 − μ) · r − rE

‖r − rE‖3

− μ · r − rM

‖r − rM‖3+ ω2

SRz(β)Rx(i)AS

− Rz(β)Rx(i)Rz(−θS)mS� − �S

‖� − �S‖3, (4)

where ωs (= 0.0748 in the time unit normalization) is theangular velocity of the Earth–Moon system with respect tothe inertial reference system, and ms (= 328900.54) is theSun’s mass in the mass unit normalization, and the positionvectors of the Earth and Moon in SE/M can be expressed asrE = [−μ 0 0]T and rM = [1 − μ 0 0]T .

Both the CR3BP and the SBCM model are classifiedas conservative Hamiltonian systems without any externalforces. Thus, the Newtonian dynamics can be deduced fromthe Hamiltonian function H1, or equivalently as

{r = ∂H1

∂p ,

p = − ∂H1∂r ,

(5)

where p = [px py pz]T is the generalized momentum,defined by the position and velocity vectors r and r as

⎧⎨

⎩

px = x − y,

py = y + x,

pz = z.

(6)

Thus the Hamiltonian function H1 can be resolved fromEqs. (4) and (5):

H1 = H0 − mS

‖� − �S‖− ω2

S · ATS · Rx(−i) · Rz(−β) · r + mS

aS, (7)

where H0 is the Hamiltonian function modeling the dynam-ics of CR3BP, and the other terms are considered as a per-turbation from the solar gravity, and the term ‖� − �S‖ can

be reproduced as

‖� − �S‖ = ∥∥aS [1 0 0]T + Rz(θs)Rx(−i)Rz(−β)r

∥∥

=√

a2S + rT r + 2aS · rC (8)

where rC is defined as rC = [cos θS sin θS 0] · Rx(−i)

Rz(−β)r, which has the same order of magnitude as ‖r‖,but is smaller than ‖r‖ (i.e., rC ≤ ‖r‖).

Moreover, H0, which can be found in textbooks, has thegeneral form

H0 = 1

2

(p2

x + p2y + p2

z

) − xpy + ypx

− 1 − μ

‖r − rE‖ − μ

‖r − rM‖ . (9)

The solar gravity leads to a periodic perturbation onto thedynamics of CR3BP, which can be characterized by the dif-ference between the two Hamiltonian functions. For a space-craft flying inside the Earth–Moon system with its distance‖r‖ from the system barycenter much shorter than aS, i.e.,‖r‖ � aS, the difference �H has the following expression:

�H = mS

[1

aS− 1

‖� − �S‖]

− ω2SAT

S Rx(−i)Rz(−β)r,

(10)

where the second term can be simplified as

1

‖� − �S‖ = 1

aS

(

1 + 2rC

aS+ ‖r‖2

a2S

)− 12

= 1

aS

[

1 − ‖r‖2

2a2S

− rC

aS+ 3

2· r2

C

a2S

+ O

(‖r‖3

a3S

)]

,

(11)

and the third term can be simplified, by Kepler’s third law,as

ω2S · aS = mS

a2S

. (12)

Therefore, the difference �H can be specified as

�H = mS

a2S

[‖r‖2

2aS− 3

2· r2

C

a3S

+ O

(‖r‖3

a4S

)]

= mS

a2S

O

(‖r‖2

aS

)

.

(13)

For the trajectories inside the Earth–Moon system discussedin this paper, the magnitude of r is close to 1 according tothe length unit normalization, i.e., ‖r‖ ≈ 1. Hence, for thehalo orbits employed in this paper, the following fact canbe obtained from their Hamiltonian values of H0 (≈ −1.6)given in Sect. 2.2: �H/H0 is of the order of magnitudeof 10−3, which can be considered as a small perturbationto the Hamiltonian system H0.


Fig. 2 Time-dependent Hill’s boundaries and equivalent equilibria: (a) the equivalent LL1 point and its Hill boundary; (b) the equivalent LL2point and its Hill boundary; the initial lunar phasic angle at the epoch time (t = 0) is θs0 = 0◦

Fig. 3 The relationship between the location of equivalent equilibriumand β: (a) for the equivalent LL1 point case, the location varies from0.836 to 0.8374 LE-M; (b) for the equivalent LL2 point case, the loca-tion varies from 1.1535 to 1.1565 LE-M; the equivalent cislunar LL1

or LL2 point is denoted as SBCM-LL1 or SBCM-LL2, to be comparedwith CR3BP-LL1 or CR3BP-LL2 in this unperturbed model; the initiallunar phasic angle at the epoch time (t = 0) is θs0 = 0◦

2.2 The definition of equivalent libration points in SBCM

The SBCM dynamics is time-dependent due to the peri-odic perturbation from the solar gravitation, compared tothe time-independent CR3BP dynamics (Koon et al. 2007;Belbruno 2004). Consequently, there are no equilibriumpoints existing in this gravitational field. Nevertheless, thegravitational equivalent equilibria will be defined in this sec-tion according to the geometry of the instantaneous Hillboundary.

For the trajectories flying inside the Earth–Moon system,the instantaneous Hill region defined by the Hamiltonianfunction H1 has a similar geometry to the constant Hamil-

tonian function H0. From the geometrical point of view,the LL1 point is essentially the critical point connecting thetwo gravitational fields around the Earth and Moon, whilethe LL2 point is the critical point connecting the interiorand the forbidden regions. Therefore, the equivalent cislu-nar LL1 point and translunar LL2 point are defined, respec-tively, as the geometrical critical points of the instantaneousHill boundary for a specified solar phasic angle β , given by[xLL1 0 0]T and [xLL2 0 0]T respectively.

Mathematically, the procedure to compute xLL1 and xLL2

is demonstrated as follows. The Hamiltonian function is anintegral of motion written in position and velocity form for-mulated by Eq. (7), and its potential function with only the


Fig. 4 The relationship between the Hamiltonian value of equivalentequilibrium and β: (a) for the equivalent LL1 point case, the Hamil-tonian value varies from −1.599 to −1.592, to be compared with theconstant value of −1.594 in the CR3BP model; (b) for the equivalent

LL2 point case, the Hamiltonian value varies from −1.594 to −1.582,to be compared with the constant value of −1.586 in CR3BP model;the initial lunar phasic angle at the epoch time (t = 0) is θs0 = 0◦

position term is formulated as

U1 = H1|r=0

= −1

2

(x2 + y2) − 1 − μ

‖r − rE‖ − μ

‖r − rM‖ − mS

‖� − �S‖− ω2

S · ATS · Rx(−i)Rz(−β) · r + mS

aS. (14)

Moreover, the Hill boundary is dominated by the potentialfunction with its velocity r = 0, known also as the zerovelocity surface. According to the definition of equivalentequilibrium mentioned above, the locations of the LL1 andLL2 points can be solved from the partial derivative of U1

with respect to the x component, i.e., ∂U1∂x

.The geometry of the time-dependent Hill boundaries,

the locations of equivalent equilibria, and their Hamiltonianvalues are, respectively, shown in Figs. 2, 3 and 4, wherethe equivalent cislunar LL1 or LL2 point is denoted asSBCM-LL1 or SBCM-LL2, to be compared with CR3BP-LL1

or CR3BP-LL2 in this unperturbed model. Due to the so-lar perturbation, the locations and the Hamiltonian valuesof equivalent equilibria are dependent on β , i.e., xLLi

=xLLi

(β), i = 1,2 and HLLi

1 = HLLi

1 (β), i = 1,2. Thus, theinitial lunar phasic angle at the epoch time (t = 0) is set asθs0 = 0◦ to produce these figures, and the solar phasic angleβ ranges from 0◦ to 360◦.

From the figures above, the location of the LL1 pointvaries from 0.836 to 0.8374 LE-M, and that of the LL2 pointvaries from 1.1535 to 1.1565 LE-M. For the equivalent LL1

point case, the Hamiltonian value varies from −1.599 to−1.592, to be compared with the constant value of −1.594in the CR3BP model; while, for the equivalent LL2 point

case, the Hamiltonian value varies from −1.594 to −1.582,to be compared with the constant value of −1.586 in theCR3BP model.

2.3 Poincaré map for lunar captured trajectories

Villac and Scheeres investigated escaping trajectories in theHill three-body problem and concluded that the decelerationat the periapsis can reach the minimum energy for the tran-siting trajectories from the LL1 or LL2 point to the Moon(Villac and Scheeres 2002). Moreover, the impulse maneu-ver �V to decelerate the spacecraft on a lunar circular orbitcan be estimated by the Hamiltonian value H1 and the ra-dius of the periapsis rp of the targeting orbit (being equalto the sum of the radius of lunar surface and the altitude ofperiapsis), as follows (Mengali and Quarta 2005):

�V (rp,H1)

= −√

μ

rp

+√

(1 − μ)2 + r2p + 2μ

rp+ 2(1 − μ)

1 − rp− 2(1 − μ)rp + H1.

(15)

However, for a specified transiting trajectory, H1 and rp

are dependent on each other, and their relationship will beinvestigated by a Poincaré map in the following section.

The technique of the Poincaré map is employed toinvestigate preliminarily the statistical features of the fuelcost and captured elements rather than a specified Moon-captured segment transiting the LL1 point, and the similarcase can be implemented for the LL2 point.


Fig. 5 Example of a Poincaré map parameterized by the Cartesiancoordinates for �H1 = 3.5 × 10−3 and β = 0◦ for LL1 point case: allthe captured trajectories are refined as the sections of Σa and Σb . Σb

is confined to remain close to the LL1 point and corresponds to thecenter manifolds of halo or Lissajous orbits near the LL1 point; Σa isconfined to remain close to the Moon and corresponds to the unstablemanifolds of halo or Lissajous orbits; only the transiting trajectoriescorresponding to Σa are discussed in this paper as the main topic onEarth-to-Moon transfers; the initial lunar and solar phasic angles at theepoch time (t = 0) are θs0 = 0◦ and β = 0◦, and the Hamiltonian valueof 300 sampling points is set as �H1 = 3.5 × 10−3

Mathematically, the procedure to compute the Poincarémap can be presented as follows. For any initial value β0

at the epoch time, two sections, Σ1 and Σ2, are used to

define the following Poincaré mapping, where the orientedsection Σ1 is located on the hyper-surface x = xLL1 with theHamiltonian flow defined by Eq. (4) from left to right:

ΣLL11 : x = xLL1(β), x > 0. (16)

Thus, all the transiting trajectories crossing this section,and dominated by the identical Hamiltonian value H1

and Eq. (7), are parameterized by the remaining four-dimensional coordinates. In this paper, this parameterizationis implemented by y0 and z0, and two direction angles (δ,φ)

ranging within the interval [−π/2,π/2] of the velocity vec-tor, whose magnitude v0 is determined by the Hamiltonianvalue H1. Therefore, the initial conditions on Σ1 can bewritten as⎧⎨

⎩

x|t=0 = xLL1, x|t=0 = v0 cos θ cos δ,

y|t=0 = y0, y|t=0 = v0 cos θ sin δ,

z|t=0 = z0, z|t=0 = v0 sin θ.

(17)

The procedure to produce the initial conditions is as follows.(i) Based on the restriction by Eq. (7), refine the interval[ymin, ymax] for the variable y0 from the case that z and v areset to be zero temporarily, and then select randomly a valueof y0 from its interval; (ii) refine the interval [zmin, zmax] for

the variable z0 from the case that v is set to be zero temporar-ily, and then select randomly a value of z0 from its interval;(iii) calculate the rest variable v0 from Eq. (7) once y0 andz0 are chosen in the steps mentioned above, i.e.,

v =√

(x2

LL1+ y2

0

) + 2

(

H1 + 1 − μ

‖r − rE‖ + μ

‖r − rM‖ + mS

‖� − �S‖ + ω2S · AT

S · Rx(−i) · Rz(−β) · r − mS

aS

)

; (18)

(iv) select randomly the direction angles δ and φ from theirinterval [−π/2,π/2].

In consequence, the initial conditions given by the sec-tion Σ1 are integrated forwards until the second section Σ2

defined as

Σ2 : rM = 0, rM > 0 (19)

where rM is the distance between the Hamiltonian flow andthe Moon, which illustrates that the section Σ2 terminatesthe integration routine at the periapsis of the integrated tra-jectory. Moreover, we define a new position vector from theflow to the Moon, as

r = [x + μ − 1 y z]T (20)

and then rewrite Eq. (17) equivalently, as

Σ2 : rT r = 0, rT r + rT a − (rT r)(rT r)rT r

> 0 (21)

where a is the acceleration in the rotating frame. In thenumerical computations performed, all the four-dimensionalcoordinates (y0, z0, θ, δ) are selected independently in thefeasible areas, and each of the coordinates involves 300 ran-dom points.

The Poincaré map defined by the flow between the twosections, i.e., Σ1 → Σ2, gives a mapping relating all thetransiting trajectories from the region near the LL1 pointdefined by the section Σ

LL11 forward to their first periapsis

defined by the section Σ2. Due to the dimensional reduc-tion by Poincaré mapping, all the captured trajectories arerefined as the sections of Σa and Σb , shown as in Fig. 5 for�H1 = 3.5 × 10−3 and β = 0◦. Essentially, Σb is confinedto remain close to the LL1 point and corresponds to the cen-ter manifolds of halo or Lissajous orbits near the LL1 point,


Fig. 6 Series of Poincaré map Σa as a function of �H1 and β forLL1 point case: the circles illustrate the lunar surface, and the chaoticpoints illustrate all the captured trajectories mapped numerically fromrandom points selected on the section Σ

LL11 ; the Hamiltonian value

�H1 has many more effects on the extrema than the solar phasic an-gle β , verified by the Poincaré map in Fig. 8; the initial lunar and solar

phasic angles at the epoch time (t0 = 0) are θs0 = 0◦ and β = 0◦, andthe subgraphs in a row have the same solar phasic angle ranked by0◦, 90◦, 180◦ and 270◦, and the subgraphs in a column have the sameHamiltonian value �H1 ranked by 5 × 10−6, 1 × 10−4, 1 × 10−3 and5 × 10−3

and Σa is confined to remain close to the Moon and cor-responds to the unstable manifolds of halo or Lissajous or-bits. Only the transiting trajectories corresponding to Σa arediscussed in this paper as the main topic on Earth-to-Moontransfers.

The consequence of all the trajectories correspondingto Σa could be deduced by their parameterization on theCartesian coordinates or classical orbital elements, whichwill be presented in the following figures. The series ofPoincaré map Σa are illustrated in the following 16 sub-graphs of Figs. 6 and 7 as a function of the Hamiltonianvalue �H1 and the solar phasic angle β , where the cir-cles illustrate the lunar surface. The chaotic points illus-trate all the captured trajectories mapped numerically from300×300×300×300 random points selected on the sectionΣ

LL11 or Σ

LL21 .

The initial conditions are listed as follows: for Fig. 5, theinitial lunar and solar phasic angles at the epoch time (t = 0)are θs0 = 0◦ and β = 0◦, respectively, and the Hamiltonianvalue of 300 sampling points is set as �H1 = 3.5 × 10−3.For Figs. 6 and 7, the initial lunar and solar phasic anglesat the epoch time (t0 = 0) are θs0 = 0◦ and β = 0◦, and thesubgraphs in a row have the same solar phasic angle rankedby 0◦, 90◦, 180◦ and 270◦, and the subgraphs in a columnhave the same Hamiltonian value �H1 ranked by 5 × 10−6,1 × 10−4, 1 × 10−3 and 5 × 10−3.

Compared with the chaotic points located on the righthand of the Moon in the rotating SE-M frame for case ofthe LL1 point, the chaotic points are located on the left handfor the case of the LL2 point, because all the captured trajec-tories reach their first periapsis on the opposite hand of theinitial section that was left, Σ

LL11 or Σ

LL21 . The fact above


Fig. 7 Series of Poincaré map Σa as a function of �H1 and β forLL2 point case: the circles illustrate the lunar surface, and the chaoticpoints illustrate all the captured trajectories mapped numerically fromrandom points selected on the section Σ

LL21 ; the Hamiltonian value

�H1 has many more effects on the extrema than the solar phasic an-gle β as well as the case of the LL1 point; the initial lunar and solar

phasic angles at the epoch time (t0 = 0) are θs0 = 0◦ and β = 0◦, andthe subgraphs in a row have the same solar phasic angle ranked by0◦, 90◦, 180◦ and 270◦, and the subgraphs in a column have the sameHamiltonian value �H1 ranked by 5 × 10−6, 1 × 10−4, 1 × 10−3 and5 × 10−3

is in accordance with the theory of Keplerian hyperbolic orparabolic orbit that the periapsis of the transfer trajectory islocated at the opposite hand of the initial leaving velocity atinfinity V∞ relative to the targeting planet.

The characteristics of the Poincaré maps vary as �H1

and β vary. They are captured by the extremum surfaces ofthe altitude of periapsis and eccentricity for all the transit-ing trajectories in Figs. 8 and 9. Inherited from the Poincarémap in Figs. 6 and 7, the Hamiltonian value �H1 has manymore effects on the extremum than the solar phasic angle β .Moreover, the maximum and minimum are illustrated, re-spectively, by the top and bottom branches of the extremumsurface, and any altitude of periapsis or eccentricity insidethe two branches is available for a specified captured trajec-

tory, demonstrated in shallow-painted areas in the projectionsubgraphs b and d of Figs. 8 and 9. In particular, the max-imum and minimum will be equal at some specified valuesof �H1 and β when the top and bottom branches join oneanother smoothly at the left edge of the extremum surface.By the subgraphs b and d, it is verified that the Hamiltonianvalue �H1 has many more effects on the extremum thanthe solar phasic angle β inherited from the Poincaré map inFigs. 6 and 7.

The procedure to produce the characteristics of the al-titude of periapsis and eccentricity of captured trajecto-ries is presented as follows: (i) collect the position andvelocity (r, r) of all the captured trajectories at their firstperiapsis based on the procedure to produce Figs. 5, 6,


Fig. 8 Extremum surfaces of altitude of periapsis and eccentricityfor all the captured trajectories for LL1 point case: (a) 3-D illustra-tion of extremum surface of altitude of the periapsis rp expressedas a function of �H1 and β; (b) 2-D projection onto the (�H1, rp)

space of the extremum; (c) 3-D illustration of extremum surface ofeccentricity e expressed as a function of �H1 and β; (d) 2-D pro-jection onto the (�H1, e) space of the extremum. The Hamiltonianvalue �H1 has many more effects on the extrema than the solar pha-sic angle β , inherited from the Poincaré map in Fig. 6; the extremumincludes both the maximum and minimum, which are illustrated, re-

spectively, by the top and bottom branches of the extremum sur-face; any altitude of periapsis or eccentricity inside the two branchesis available for a specified transiting trajectory, demonstrated in theshallow-painted areas in the 2-D illustrations b and d. The maxi-mum and minimum are equal to each other at some specified val-ues of �H1 and β when the top and bottom branches join one an-other smoothly at the left edge of the extremum surface; the ini-tial lunar phasic angle is θs0 = 0◦, the solar phasic angle β rangesfrom 0◦ to 360◦, and the Hamiltonian value �H1 ranges from 0 to5 × 10−3

and 7; (ii) transform the state (r, r) from the syzygy SE-M

frame to the Moon-center inertial frame, which has thesame coordinate axis definition as that of the inertial frameIS-E/M or IE-M, but fixes its origin at the barycenter ofthe Earth–Moon system; (iii) convert the updated inertialstates into the classical orbital elements, including rp and e,based on the lunar gravitational coefficients and Keple-rian two-body theory (the conversion between classical or-bital elements and Cartesian coordinates is common andcan be found in textbooks); (iv) plot the extremum sur-faces in Figs. 8 and 9 for the LL1 and LL2 points, respec-tively.

The initial conditions are listed as follows: the initial lu-nar phasic angle is θs0 = 0◦, and the solar phasic angle β

ranges from 0◦ to 360◦, and the Hamiltonian value �H1

ranges from 0 to 5 × 10−3 (for the LL1 point) or from 0 to1.1 × 10−3 (for the LL2 point).

A tangential burn �V is required to capture a circular or-bit about the Moon, which is also regarded as the criterion tomeasure some candidate trajectories from the viewpoint ofenergy. Considering a fixed radius of periapsis captured bythe Moon, e.g., rp = 1838 km (i.e., the altitude of periapsisis equal to 100 km), only a specified Hamiltonian value �H1

is refined from the shallow-painted areas shown in Figs. 10a


Fig. 9 Extremum surfaces of altitude of periapsis and eccentricity forall the captured trajectories for LL2 point case: (a) 3-D illustrationof extremum surface of altitude of periapsis rp expressed as a func-tion of �H1 and β; (b) 2-D projection onto the (�H1, rp) space ofthe extremum; (c) 3-D presentation of extremum surface of eccen-tricity e expressed as a function of �H1 and β; (d) 2-D projectiononto the (�H1, e) space of the extremum; the Hamiltonian value�H1 has many more effects on the extrema than the solar phasicangle β , inherited from the Poincaré map in Fig. 7. The extremumincludes both the maximum and minimum, which are illustrated,

respectively, by the top and bottom branches of the extremum sur-face; any altitude of periapsis or eccentricity inside the two branchesis available for a specified transiting trajectory, demonstrated in theshallow-painted areas in the 2-D illustrations (b) and (d). The maxi-mum and minimum are equal to each other at some specified valuesof �H1 and β when the top and bottom branches join one anothersmoothly at the left edge of the extremum surface. The initial lunarphasic angle is θs0 = 0◦, the solar phasic angle β ranges from 0◦ to360◦, and the Hamiltonian value �H1 ranges from 0 to 1.1 × 10−3

and 10b for an arbitrary β ∈ [0,2π], and then the minimum�Vmin can be obtained from Eq. (14) by the refined mini-

mum value of �H1. Thus, the improved Poincaré mappingwith a fixed rp establishes the relationship between �Vmin

and β , as illustrated in Fig. 10. Compared with the cap-tured �Vmin of 695.7 m/s yielded by the Keplerian two-body

model, 656.8 m/s by the Hill model, and 649.2 m/s (LL1

point) and 652.9 m/s (LL2 point) in the CR3BP model (Vil-lac and Scheeres 2002; He and Xu 2007), the SBCM model

can reach the minimum value of 642.9 m/s (LL1 point) and646.7 m/s (LL2 point).

Mathematically, the procedures to produce Figs. 8 and 9

establish a mapping (or function) from H1 and β to rp,

which is formulated as rp = Γ (H1, β); however, for a fixed

radius of periapsis r∗p = 1838 km, H1 is parameterized by

the only variable, β , i.e., H1 = Γ −1r∗p

(β), which can be solved

from numerical procedures of Figs. 8 and 9. Subsequently,

the minimum �Vmin can be obtained from Eq. (14) by the

refined minimum value of H1. The initial conditions to pro-

duce Fig. 10 are listed as follows: the initial lunar phasic

angle is θs0 = 0◦, and the solar phasic angle β ranges from

0◦ to 360◦.


Fig. 10 The relationship between the minimal captured �Vmin andβ produced by the improved Poincaré mapping with a fixed altitudeof periapsis rp = 1838 km: (a) the relationship for LL1 point case;(b) the relationship for the case of the LL2 point; compared withthe captured �Vmin of 695.7 m/s yielded by the Keplerian two-body

model, 656.8 m/s by the Hill model, and 649.2 m/s (LL1 point) and652.9 m/s (LL2 point) in the CR3BP model (Villac and Scheeres 2002;He and Xu 2007), the SBCM model can reach the minimal value of642.9 m/s (LL1 point) and 646.7 m/s (LL2 point)

Fig. 11 Cislunar transfer opportunities measured by the solar pha-sic angle β: the solid lines illustrate the available altitudes of peri-apsis about the Earth, and the dash-dotted lines illustrate the avail-able altitudes of periapsis about the Moon. The intersection of theintervals transiting from the Earth to the LL1 point and from theLL1 point to the Moon is bounded by the vertical dashed lines, i.e.,[77◦,109◦] ∪ [285◦,342◦], which drives the trajectories to orbit suc-cessively the Earth and Moon. The gaps between the altitudes of peri-apsis are caused by some β intervals with failing invariant manifoldsin transiting the LL1 point

3 Low-energy transfers by transiting equivalentlibration points

Compared with the statistical features of captured orbital el-ements discussed in the section above, the minimum-energy

cislunar and translunar trajectories are yielded by transit-ing the LL1 and LL2 points in this section. It is studiedhow the asymptotical behaviors of invariant manifolds ap-proaching to or recessing from the libration points or haloorbits are destroyed by the solar perturbation. Moreover, thetransfer opportunities measured by the solar phasic angle β

are achieved for the Earth-escaping and Moon-captured seg-ments, respectively.

3.1 Low-energy transfers by transiting the LL1 point

For the CR3BP model, the minimum-energy trajectory tran-siting the LL1 point is essentially two branches of the invari-ant manifolds originating from this equilibrium point. Thus,the spacecraft may follow the stable manifold from the inte-rior region dominated by the Earth’s gravity to the LL1 point,and then leave along the unstable manifold for the exteriorregion dominated by the lunar gravity. However, this trans-fer trajectory is not practical because its duration is infinite,which is inherited from the fact that the invariant manifoldsapproach or leave the LL1 point asymptotically in an infiniteduration.

The perturbation of the solar gravity employed by theSBCM model will change topologically the invariant man-ifolds to fail in transiting the LL1 point for some phasicangles β ∈ [0,2π]; however, the transiting manifolds arepreserved for the other values of β , inheriting from the time-invariant CR3BP model. For the interval of β transiting theLL1 point, the asymptotical infinite durations are cut downto finite ones by the perturbation, which is quite beneficial toEarth-to-Moon transfers. For the interval of β not transiting


Fig. 12 Typical cislunartransfer trajectory transiting theLL1 point in the rotating SE-Mframe for β = 286◦: the bluesolid lines illustrate the segmentescaping from the Earth, and thered dash-dotted lines illustratethe segment captured by theMoon. The z component rangesbetween −2 × 10−3 ∼+2 × 10−3 (LE-M), while the x

and y components range,respectively, between−0.9 ∼ +1.2 and −0.8 ∼ +0.8(LE-M), which indicates that thespatial perturbation has feweffects on the low-energytransfer

Fig. 13 Typical cislunartransfer trajectory transiting theLL1 point in the inertial IE-Mframe for β = 286◦: the bluesolid lines illustrate the segmentescaping from the Earth, and thered dash-dotted lines illustratethe segment captured by theMoon. The z component rangesbetween −2 × 10−3 ∼+2 × 10−3 (LE-M), while the x

and y components range,respectively, between−0.9 ∼ +1.2 and −0.8 ∼ +0.8(LE-M). This indicates that thespatial perturbation has feweffects on the low-energytransfer

the LL1 point, the perturbed manifolds will lose the phase oforbiting the Earth or Moon, i.e., there is no periapsis aboutthe Earth or Moon in this case. Therefore, the gaps betweenthe altitudes of periapsis about the Earth and Moon are pre-sented by the phasic angle β in Fig. 11. Only the intersec-tion between β intervals transiting from the Earth to the LL1

point and other intervals transiting from the LL1 point to theMoon, i.e., [77◦,109◦] ∪ [285◦,342◦], can drive the trajec-tories to orbit, successively, the Earth and Moon, and can bealso considered as the cislunar transfer opportunities; theyare bounded by the vertical dashed lines in Fig. 11.

The procedure to produce the cislunar transfer oppor-tunities measured by the solar phasic angle β is presentedas follows. Vary β in the interval of [0◦,360◦] to integratebackwards the SBCM dynamics formulated by the differen-tial Eq. (4) backwards to yield the transfer opportunities forEarth-escaping segment, and integrate forwards to yield the

transfer opportunities for Moon-captured segment. The twointegrations (forwards and backwards) have the same ini-tial condition of [xLL1 ,0,0,0,0,0]T . Only particular subin-tervals of β can make the integrated trajectories closer tothe Earth or Moon; it is those that are considered as cis-lunar or translunar transfer opportunities. The initial condi-tions are: the initial lunar phasic angle is θs0 = 0◦, the solarphasic angle β ranges from 0◦ to 360◦, and the initial val-ues to integrate forwards and backwards Eq. (4) are equally[xLL1,0,0,0,0,0]T .

The three-dimensional cislunar trajectory is presentedin SE-M (Fig. 12) and IE-M frames (Fig. 13). From thetwo figures, it is deduced that the z component ranges be-tween −2 × 10−3 ∼ +2 × 10−3 (LE-M), while the x and y

components range, respectively, between −0.9 ∼ +1.2 and−0.8 ∼ +0.8 (LE-M). This conclusion can also be summa-rized from the trajectories transiting the LL2 point or halo


Fig. 14 Osculating semi-major axis and eccentricity of a cislunartransfer trajectory for β = 286◦: (a) the history of the osculating semi-major axis; (b) the history of the osculating eccentricity. The dash-dotted lines illustrate the elements during escaping from the Earth, andthe dashed lines illustrate the elements during being captured by theMoon, and the vertical solid lines indicate the time epoch (t = 0); theosculating semi-major axis and eccentricity before the epoch (t < 0)are converted from the position and velocity in the Earth-center in-ertial frame, and the osculating elements after the epoch (t < 0) are

converted in the Moon-center inertial frame; compared with the lu-nar perturbation before the epoch, the Earth has more perturbation onthe osculating orbital elements converted in the Moon-center inertialframe. This cislunar transiting trajectory is classified as low-energytransfer because the eccentricities before and after the epoch are lessthan 1, compared to the hyperbolical velocity captured by the Moonin classical Hohmann transfer (like the Apollo (NASA) and Chang’E(China) missions)

orbits near the two libration points. Thus, for all the cislu-nar and translunar trajectories discussed in this paper, the z

component is much smaller than the other components (thez component is only about one thousandth of the x or y com-ponent), which indicates that the spatial perturbation has feweffects on the low-energy transfer.

The procedure to produce a typical cislunar transfer tra-jectory transiting the LL1 point in the rotating SE-M frame is:for some specified value of β , integrate Eq. (4) backwardsfrom the equivalent equilibrium to obtain the Earth-escapingsegment and forwards to achieve the Moon-captured seg-ment in the rotating SE-M frame. The transfer trajectories RI

in the inertial IE-M frame in Fig. 13 are converted from theintegrated trajectories r in the rotating SE-M frame in Fig. 12,based on the transition matrix of RI = Rx(−i)Rz(−β)r.The initial conditions are: the initial lunar phasic angle atthe epoch time (t = 0) is θs0 = 0◦ for the four figures, andthe initial solar phasic angle β at the epoch time (t = 0) is286◦ for Figs. 12 and 13. The integral initial values to pro-duce Figs. 12 and 13 are [xLL1 ,0,0,0,0,0]T .

The semi-major axis and eccentricity of the typical cis-lunar low-energy trajectory for β = 286◦ are illustrated inFig. 14. Due to the solar gravitational perturbation, theduration of the transiting manifold is finite. For a cislunartrajectory, the time epoch (t = 0) is set as the moment ofpassing through the LL1 point, and its stable manifold willorbit the Earth before this epoch (i.e., t < 0), while its un-stable manifold will orbit the Moon after this epoch (i.e.,

Fig. 15 WSB Transfer opportunities by transiting the LL2 point mea-sured by the solar phasic angle β: the solid lines illustrate the availablealtitudes of periapsis about the Earth, and the dash-dotted lines illus-trate the available altitudes of periapsis about the Moon; the intersec-tion of the intervals transiting from the Earth to the LL1 point and fromthe LL1 point to the Moon is bounded by the vertical dashed lines, i.e.,[21.8◦,23.3◦] ∪ [201.5◦,203◦], which drives the trajectories to orbitsuccessively the Earth and Moon; the gaps between the altitudes of pe-riapsis are caused by some intervals with failing invariant manifolds intransiting the LL2 point

t > 0). Thus, the osculating semi-major axis and eccentricitybefore the epoch (t < 0) should be converted from the posi-tion and velocity in the Earth-center inertial frame based on


the Keplerian restricted two-body theory, while the osculat-ing semi-major axis and eccentricity after the epoch (t > 0)should be converted in the Moon-center inertial frame. Inthis case, the jumps at the epoch (t = 0) are caused by thefact that the orbital elements before and after this epoch areconverted in two different inertial frames, i.e., the former isin the Earth-center frame but the latter is in the Moon-centerone.

Furthermore, the orbital elements after the epoch havea more considerable variation in amplitude than before theepoch, especially for the eccentricity. It is because the oscu-lating orbital elements are achieved only based on the Ke-plerian restricted two-body theory; however, the perturba-tion from the other celestial body’s gravity affects the ele-ments greatly, which is referred to as the second body in theCR3BP or SBCM model. Compared with the lunar perturba-tion before the epoch, the Earth has more perturbation on theosculating orbital elements converted in the Moon-center in-

ertial frame after the epoch, which accounts for more jumpson the eccentricity (in the right subgraph of Fig. 14) afterthe epoch than before the epoch. This cislunar transitingtrajectory is classified as low-energy transfer because boththe eccentricities before and after the epoch are less than 1,compared to the hyperbolical velocity captured by the Moonin classical Hohmann transfer (like the Apollo (NASA) andChang’E (China) missions).

The procedure to produce Fig. 14 is: (i) integrate Eq. (4)backwards to obtain the Earth-escaping segment and for-wards to achieve the Moon-captured segment in the rotat-ing SE-M frame, both from the same initial condition of[xLL1,0,0,0,0,0]T ; (ii) transform the state (r, r) from thesyzygy SE-M frame to the Earth-center inertial frame for theEarth-escaping segment, and transform the state (r, r) fromthe syzygy SE-M frame to the Moon-center inertial framefor the Moon-captured segment; (iii) convert the semi-majoraxis a and eccentricity e before the epoch (t < 0) from the

Fig. 16 Typical innertranslunar transfer trajectorytransiting the LL2 point in therotating SE-M frame forβ = 193◦: the blue solid linesillustrate the segment escapingfrom the Earth, and the reddash-dotted lines illustrate thesegment captured by the Moon;the z component ranges between−3 × 10−3 ∼ +3 × 10−3

(LE-M), while the x and y

components range between−0.9 ∼ +1.2 and −0.8 ∼ +0.8(LE-M). This indicates that thespatial perturbation has feweffects on the low-energytransfer

Fig. 17 Typical innertranslunar transfer trajectorytransiting the LL2 point in theinertial IE-M frame forβ = 193◦: the blue solid linesillustrate the segment escapingfrom the Earth, and the reddash-dotted lines illustrate thesegment captured by the Moon;the z component ranges between−3 × 10−3 ∼ +3 × 10−3

(LE-M), while the x and y

components range between−0.9 ∼ +1.2 and −0.8 ∼ +0.8(LE-M). This indicates that thespatial perturbation has fewereffects on the low-energytransfer


Earth-escaping segment based on the Earth’s gravitationalcoefficients, and convert a and e after the epoch (t > 0) fromthe Moon-captured segment based on the lunar gravitationalcoefficients. The initial conditions are: the initial lunar pha-sic angle at the epoch time (t = 0) is θs0 = 0◦ for the fourfigures; the initial solar phasic angle β at the epoch time(t = 0) is 286◦. The integral initial values to produce Fig. 14are [xLL1,0,0,0,0,0]T .

The low-energy cislunar transfers transiting the LL1 pointhave minimum energy, because the LL1 point has minimumenergy in itself compared to the LL2 point and periodic or-bits near the two equivalent equilibria.

3.2 Low-energy transfers by transiting the LL2 point

Similar to transiting the LL1 point, the solar perturbationwill change topologically the invariant manifolds failing in

transiting the LL2 point for some phasic angles β ∈ [0,2π];however, the transiting manifolds are preserved for the othervalues of β . For the available interval of β , the translunartransfer opportunities for the LL2 point can be produced bythe procedure developed for the LL1 point. The initial con-ditions are: the initial lunar phasic angle is θs0 = 0◦, and thesolar phasic angle β ranges from 0◦ to 360◦, and the ini-tial values to integrate forwards and backwards Eq. (4) areequally [xLL2,0,0,0,0,0]T . The translunar transfer oppor-tunities are illustrated in Fig. 15.

Different from the only type of cislunar trajectories tran-siting the LL1 point, all the transfer trajectories transitingthe LL2 point are classified as the inner cislunar trajectoriesand the outer WSB translunar ones. The former is essen-tially the cislunar transfer trajectories passing through theLL1 point, and costs more fuel than the cislunar trajectories

Fig. 18 Typical outer translunartransfer trajectory transiting theLL2 point in the rotating SE-Mframe for β = 202◦: the bluesolid lines illustrate the segmentescaping from the Earth, and thered dash-dotted lines illustratethe segment captured by theMoon. The z component rangesbetween 0 ∼ +0.12 (LE-M),while the x and y componentsrange between −4 ∼ +4 and−4 ∼ +4 (LE-M), whichindicates that the spatialperturbation has few effects onthe low-energy transfer

Fig. 19 Typical outer translunartransfer trajectory transiting theLL2 point in the inertial IE-Mframe for β = 202◦: the bluesolid lines illustrate the segmentescaping from the Earth, and thered dash-dotted lines illustratethe segment captured by theMoon; the z component rangesbetween 0 ∼ +0.12 (LE-M),while the x and y componentsrange between −2.7 ∼ +1.3 and−3.4 ∼ +1.2 (LE-M), whichindicates that the spatialperturbation has few effects onthe low-energy transfer; thistype of transfer trajectories inthe inertial frame has the samegeometrical shape as Belbruno’sWSB theory (Belbruno andMiller 1993; Belbruno 2004),which is called outer WSBtrajectories as well


transiting the LL1 point. The latter has the same geometri-cal shape in the inertial frame as in Belbruno’s theory andis named after the outer translunar WSB trajectories in thispaper (Belbruno and Miller 1993; Belbruno 2004), whichcan be considered as the patched connection between the

invariant manifolds near the EL1 (or EL2) point and unsta-ble manifolds near the LL2 point (Koon et al. 2001). There-fore, the former is not a fuel-efficient Earth-to-Moon trans-fer, and only the translunar WSB trajectories are employedin this paper to transit the LL2 point. Due to the harsh con-

Fig. 20 Osculating semi-major axis and eccentricity of a cislunartransfer trajectory for β = 193◦: (a) the history of the osculating semi-major axis; (b) the history of the osculating eccentricity. The dash-dotted lines illustrate the elements during escaping from the Earth, andthe dashed lines illustrate the elements during captured by the Moon,and the vertical solid lines illustrate the time epoch (t = 0). The os-culating semi-major axis and eccentricity before the epoch (t < 0) areconverted from the position and velocity in the Earth-center inertial

frame, and the osculating elements after the epoch (t < 0) are con-verted in the Moon-center inertial frame; compared with the Moon be-fore the epoch, the Earth has more perturbation on the osculating or-bital elements converted in the Moon-center inertial frame. This cislu-nar transiting trajectory is classified as low-energy transfer because theeccentricities before and after the epoch are less than 1, compared tothe hyperbolical velocity captured by the Moon in classical Hohmanntransfer (like the Apollo (NASA) and Chang’E (China) missions)

Fig. 21 Osculating semi-major axis and eccentricity of a cislunartransfer trajectory for β = 202◦: (a) the history of the osculating semi-major axis; (b) the history of the osculating eccentricity. The dash-dotted lines illustrate the elements during escaping from the Earth, andthe dashed lines illustrate the elements during captured by the Moon,and the vertical solid lines illustrate the time epoch (t = 0); the osculat-ing semi-major axis and eccentricity before the epoch (t < 0) are con-verted from the position and velocity in the Earth-center inertial frame,

and the osculating elements after the epoch (t < 0) are converted inthe Moon-center inertial frame. Compared with the lunar perturbationbefore the epoch, the Earth has more perturbation on the osculating or-bital elements converted in the Moon-center inertial frame; this cislu-nar transiting trajectory is classified as low-energy transfer because theeccentricities before and after the epoch are less than 1, compared withthe hyperbolical velocity captured by the Moon in classical Hohmanntransfer (like the Apollo (NASA) and Chang’E (China) missions)


ditions of two patched manifolds, only a few of the intervalscan be used to construct the whole WSB transfer trajecto-ries from the Earth to the LL1 point and then to the Moon,which are bounded by the vertical dashed lines in Fig. 15,i.e., β ∈ [21.8◦,23.3◦] ∪ [201.5◦,203◦]. Compared to thecislunar transfer opportunities listed in Fig. 11, the WSBtransfers have fewer opportunities to transit the LL2 point.

The procedures to produce transfer trajectories transitingthe LL1 point in the rotating SE-M frame and the inertial IE-M

Fig. 22 Halo orbits near LL1 and LL2 points and serial points on theorbits: the maximal values of the y components of halo orbits are, re-spectively, ±40142.16 km near the LL1 point and ±33818.07 km nearthe LL2 point; the series of evenly spaced points in time are selectedclockwise from the starting point; the starting point is located closestto the Earth on the x axis

frame can be employed to produce the two types of innertranslunar and outer translunar transfer trajectories transit-ing the LL2 point, as shown in Figs. 16, 17, 18, and 19. Forthe four figures, the initial lunar phasic angle at the epochtime (t = 0) is θs0 = 0◦, and the integral initial value is[xLL2,0,0,0,0,0]T . For Figs. 16 and 17, the initial solarphasic angle β at the epoch time (t = 0) is 193◦; and forFigs. 18 and 19, the initial solar phasic angle β at the epochtime (t = 0) is 202◦.

Furthermore, the procedures to create the osculatingsemi-major axis and eccentricity for a cislunar transfer tra-jectory in Sect. 3.1 can be used to deal with the LL2 pointcase. The time history of the orbital elements is presented inFigs. 20 and 21, respectively, for typical inner translunar andouter WSB translunar transfer trajectories transiting the LL2

point. The initial lunar phasic angle and the integral initialvalue are θs0 = 0◦ and [xLL2,0,0,0,0,0]T for the two fig-ures, and the initial solar phasic angle β of Fig. 20 is 193◦,and the initial solar phasic angle β of Fig. 21 is 202◦.

The cislunar transfer trajectories transiting the LL1 pointhave a total opportunity measured by �β = 89◦, while theouter WSB translunar trajectories have much fewer opportu-nities than �β = 3◦. Thus, an effective way to increase thetransfer opportunities for the WSB trajectories is to transit ahalo orbit near the LL2 point instead of itself.

4 Low-energy transfers by transiting halo orbits

Compared with the only variable (i.e., β) to design a trans-fer trajectory transiting the libration point, the halo orbit is

Fig. 23 Invariant manifolds of a halo orbit and cislunar transfer tra-jectories constructed from these manifolds: (a) Ws

E illustrates the sta-ble branch leaving from the Earth for the halo orbit, and Wu

E illustratesthe unstable branch leaving from the halo orbit for the Earth, and Ws

Millustrates the stable branch leaving from the Moon for the halo or-

bit, and WuM illustrates the unstable branch leaving from the halo orbit

for the Moon; (b) the branches WsM and Wu

E form a whole Moon-to-Earth transfer trajectory labeled by light-colored lines, while Ws

E andWu

M form a whole Earth-to-Moon transfer trajectory labeled by dark-colored lines


employed to increase the transfer opportunities by introduc-ing another variable (i.e., serial points of halo orbit). Subse-quently, a global investigation on the Earth-escaping and theMoon-captured opportunities is implemented, respectively,for transiting the LL1 and LL2 points in this section.

A halo orbit is a periodic three-dimensional orbit nearthe LL1 and LL2 points, and is symmetrical about the x–z

plane in the rotating SE-M frame (Xu and Xu 2012), shownin Fig. 22. A halo orbit in the Earth–Moon system can becharacterized by the maximum of its y component or its or-bital period TH . Thus, all points on a specified halo orbitcan be marked by the phase of halo orbit τ = i/N , where i

is the serial number of this point measured clockwise fromthe starting point which is located closest to the Earth on thex axis, and N = 360 is the total number of evenly spacedpoints in time selected in this paper. Even though there is nohalo orbit under the solar perturbation in SBCM model, theperiodic orbit is still acting as a powerful tool to investigatethe transfer trajectories in this paper, because both the cislu-nar and the translunar trajectories are transiting it rather thanstaying on it (Koon et al. 2001, 2007). The algorithm to pro-duce a halo orbit is beyond the scope of this paper; it can befound in the references (Richardson 1980; Xu et al. 2013).The maximal values of the y components of halo orbits inFig. 22 are, respectively, ±40142.16 km near LL1 point and±33818.07 km near LL2 point.

4.1 Low-energy transfers by transiting halo obits near LL1

point

In the CR3BP model, the invariant manifolds of a halo or-bit near the LL1 point can be classified into four branchesas Ws

E, WuE , Ws

M, and WuM, where the subscript E and M

indicate that this branch leaves from/goes to the Earth andMoon, respectively, and the superscript s and u indicatethe branch approaches of the halo orbit forwards and back-wards, respectively. Thus, the branches Ws

M and WuE form

a whole Moon-to-Earth transfer trajectory labeled by light-colored lines in the right subgraph of Fig. 23, while Ws

E andWu

M form a whole Earth-to-Moon transfer trajectory labeledby dark-colored lines in the right subgraph of Fig. 23. How-ever, both of the two trajectories are not practical due to theinfinite durations. The algorithm to produce invariant man-ifolds of the halo orbit is beyond the scope of this paper; itcan be found in the references (Howell et al. 1997, 2006).The maximal values of the y components of halo orbits inthis figure are, respectively, ±40142.16 km near the LL1

point and ±33818.07 km near the LL2 point.Fortunately, the perturbation of the solar gravity em-

ployed by the SBCM model may cut down the infinite dura-tions of some branches to finite ones, which is quite prac-tical for Earth-to-Moon transfers. For the pairs (β, τ ) ∈[0,2π] × [0,1] not transiting a halo orbit near the LL1

Fig. 24 Contour-map of transfer opportunities for translunar WSBtrajectories: (a) transfer opportunities for the Earth-escaping seg-ments; (b) transfer opportunities for the Moon-captured segments;the solar phasic angle β has more effects on the existence of thetrajectories than the phase of halo orbit, and most of them arelocated within the (β, τ ) pairs of ([95◦,150◦] ∪ [262◦,345◦]) ×[0,1] (for the Earth-escaping segment) and ([100◦,200◦] ∪[270◦,30◦]) × ([0.16,0.26] ∪ [0.47,0.58]) (for the Moon-capturedsegment); all the cislunar trajectories mapped from the contour-maphave the similar geometrical shape with the typical trajectories shownin Fig. 25

point, the perturbed manifolds will lose the phase of or-biting the Earth or the Moon, i.e., there is no periapsisabout the Earth or the Moon in this case. Only the inter-secting pairs of transiting from the Earth to halo orbit andother intervals transiting from halo orbit to the Moon, i.e.,([95◦,150◦] ∪ [262◦,345◦]) × [0,1] and ([100◦,200◦] ∪[270◦,30◦])×([0.16,0.26]∪[0.47,0.58]), can drive the tra-jectories to orbit successively the Earth and Moon, which isconsidered as the cislunar transfer opportunities shown inFig. 24.

The procedure to produce the transfer opportunities ispresented as follows: (i) vary β in the interval of [0◦,360◦]


Fig. 25 Typical cislunar transfer trajectory transiting a halo obit nearthe LL1 point in the rotating SE-M frame for β = 150◦ and τ = 82/360:the blue solid lines illustrate the segment escaping from the Earth, andthe red dash-dotted lines illustrate the segment captured by the Moon,and the black thick lines illustrate the halo orbit; the x and y compo-

nents range between −0.9 ∼ +1.2 and −0.8 ∼ +0.8 (LE-M), whilethe z component ranges between −1.1 ∼ +1.1 (LE-M), which is largerthan the z component of a trajectory transiting the LL1 point; the max-imal y component of the halo obit is ±40142.16 km

and the phase of serial points on halo orbit τ ∈ [0,360]/360to integrate the SBCM dynamics formulated by the differ-ential Equation (4) backwards to yield the Earth-escapingsegment, and integrate forwards to yield the Moon-capturedsegment; (ii) collect the altitudes of periasis when the Earth-escaping or the Moon-captured segments reach their first pe-riasis, and then draw them by the contour-map of rp, whichare considered as cislunar or translunar transfer opportu-nities; (iii) the two integrations (forwards and backwards)have the same initial condition of X = [r, r]|τ of a pointof the series on the halo orbit. Only some subintervals ofβ and τ can make the integrated trajectories closer to theEarth or Moon. The initial conditions are: the initial lu-nar phasic angle is θs0 = 0◦, and the solar phasic angle β

ranges from 0◦ to 360◦, and the phase of serial points onhalo orbit τ ranges from 0 to 1. The maximal y componentof halo orbit to integrate forwards and backwards Eq. (4)is ±40142.16 km.

The procedures to produce transfer trajectories transitingthe LL1 point in the rotating SE-M frame can be employed toproduce the cislunar transfer trajectories transiting halo orbitin Fig. 25: for some specified pair (β, τ ), integrate Eq. (4)backwards from a halo orbit to obtain the Earth-escapingsegment and forwards to achieve the Moon-captured seg-ment in the rotating SE-M frame. The initial conditions arelisted as follows: the initial lunar and solar phasic angle atthe epoch time (t = 0) is θs0 = 0◦ and β = 150◦, respec-tively. The two integrations (forwards and backwards) havethe same initial condition X = [r, r]|τ=82/360 as a point ofthe series on the halo orbit with its maximal y componentequal to ±40142.16 km.

Fig. 26 The conceptual geometry of two smooth-patched manifoldson a Poincaré section: both the invariant manifolds of halo orbits nearthe EL1 and EL2 points can be used to construct the whole of thetranslunar WSB trajectories

It is worth mentioning that the numerical simulationsindicate that all the cislunar trajectories mapped from thecontour-map have a similar geometrical shape to the typicaltrajectories shown in Fig. 25.

4.2 Low-energy transfers by transiting halo obits nearthe LL2 point

Similar to transiting the LL2 point, the inner transfer tra-jectories transiting the halo orbit near the LL2 point are es-sentially the cislunar transfer trajectories passing through


LL1 point, and cost more fuel than the cislunar trajecto-ries achieved in Sect. 4.1. Hence, only the translunar WSBtrajectories are employed in this paper so as to constructsome practical transfer trajectories. According to the workof Koon et al. (2001), there are smooth-patched manifoldson a Poincaré section to drive the spacecraft flying from theEarth to another halo orbit near the EL1 (or EL2) point andthen to the targeting halo orbit near the LL2 point, which areillustrated in Fig. 26. Thus, the following investigation willverify that both the invariant manifolds of halo orbits nearEL1 and EL2 points can be used to construct the whole ofthe translunar WSB trajectories.

The procedure developed for the transfer opportunitiesin the above section can also be used to create the colorfulPorkchop-like contour-maps of transfer opportunities for theEarth-escaping and the Moon-captured segments transitingthe halo orbit near the LL1 point. The initial conditions are:the initial lunar phasic angle is θs0 = 0◦, and the solar pha-sic angle β ranges from 0◦ to 360◦, and the phase of serialpoints on halo orbit τ ranges from 0 to 1. The maximal y

component of the halo orbit to integrate forwards and back-wards Eq. (4) is ±33818.07 km.

Only the intersecting pairs transiting from the Earth tothe halo orbit and other intervals transiting from the halo or-bit to the Moon, i.e., ([85◦,165◦] ∪ [262◦,330◦]) × [0,1]and ([10◦,92◦] ∪ [192◦,268◦]) × ([0,0.26] ∪ [0.57,0.66]),can drive the trajectories to orbit successively the Earth andMoon, and is considered as a case of the cislunar trans-fer opportunities shown in Fig. 27. The points labeled asa,b, . . . ,n, are mapped into 14 typical translunar WSB tra-jectories in the rotating SS-E/M frame, which can be pro-duced by the following procedure: (i) integrate Eq. (4) back-wards to obtain the Earth-escaping segment and forwardsto achieve the Moon-captured segment in the rotating SE-M

frame both from the same initial condition of a serial pointon the halo orbit; (ii) the transfer trajectories RI in the SS-E/M

frame are converted from the integrated trajectories r inthe rotating SE-M frame, based on the transition matrix ofRI = Rx(−i)Rx(−β)r + AS.

The initial conditions are listed as follows. The initial lu-nar phasic angle at the epoch time (t = 0) is θs0 = 0◦, andthe initial condition X = [r, r]|τ is selected from a point ofthe series with its phase τ on the halo orbit. All the sub-graphs are produced by the halo orbit with its maximal y

component equal to ±33818.07 km. The initial phase ofserial points τ and the solar phasic angle β at the epochtime (t = 0) are: (a) τ = 345/360 and β = 10◦; (b) τ =350/360 and β = 33◦; (c) τ = 177/360 and β = 96◦;(d) τ = 13/360 and β = 13◦; (e) τ = 360/360 and β = 53◦;(f) τ = 352/360 and β = 35◦; (g) τ = 352/360 and β =33◦; (h) τ = 52/360 and β = 205◦; (i) τ = 24/360 and β =194◦; (j) τ = 220/360 and β = 13◦; (k) τ = 62/360 andβ = 191◦; (l) τ = 51/360 and β = 205◦; (m) τ = 187/360and β = 89◦; (n) τ = 189/360 and β = 89◦.

Fig. 27 Contour-map of transfer opportunities for translunar WSB tra-jectories: (a) transfer opportunities for the Earth-escaping segments;(b) transfer opportunities for the Moon-captured segments; the solarphasic angle β has more effects on the existence of the trajectoriesthan the phase of halo orbit, and most of them are located in the β in-tervals of ([85◦,165◦] ∪ [262◦,330◦]) × [0,1] (for the Earth-escapingsegment) and ([10◦,92◦] ∪ [192◦,268◦]) × ([0,0.26] ∪ [0.57,0.66])(for the Moon-captured segment); the phase of halo orbit has more ef-fects on the altitudes of periapsis about the Earth and Moon; the pointslabeled as a,b, . . . ,n, are mapped into 14 typical translunar WSB tra-jectories, as shown in the rotating SS-E/M frame in Fig. 28; the maximaly component of the halo obit is ±33818.07 km

Because of the conclusion in Sect. 2.2 that all the cis-lunar and translunar trajectories have a z component muchsmaller than the other components, only the x–y view is pre-sented for the labeled points, a,b, . . . ,n. The 14 typical tra-jectories are classified as the ones passing through the EL1

point (i.e., a,b, . . . ,g) and the others passing through theEL2 point (i.e., h, i, . . . ,n). Thus, both the EL1 and the EL2

points can be employed to join with the unstable manifoldsof a halo or a Lyapunov orbit near the LL2 point in driv-


Fig. 28 Typical translunar WSB trajectories in the SS-E/M frame: allthe subgraphs correspond to the points labeled in Figs. 19 and 21.The phase of the halo orbit and the solar phasic angle are selected as:(a) τ = 345/360 and β = 10◦; (b) τ = 350/360 and β = 33◦; (c)τ =177/360 and β = 96◦; (d) τ = 13/360 and β = 13◦; (e) τ = 360/360and β = 53◦; (f) τ = 352/360 and β = 35◦; (g) τ = 352/360 andβ = 33◦; (h) τ = 52/360 and β = 205◦; (i) τ = 24/360 and β = 194◦;

(j) τ = 220/360 and β = 13◦; (k) τ = 62/360 and β = 191◦; (l)τ = 51/360 and β = 205◦; (m) τ = 187/360 and β = 89◦; (n) τ =189/360 and β = 89◦. (a), (b), . . . , (g) are classified as the translunarWSB trajectories passing through the EL1 point, and (h), (i), . . . , (n)

are classified as the ones passing through the EL2 point; the circles in-dicate the lunar surface; the maximal y component of the halo obit is±33818.07 km


Fig. 28 (Continued)

ing the spacecraft from the Earth to the Moon, which is inaccordance to Koon et al.’s conclusion (Koon et al. 2001).In the SS-E/M frame, the Earth and the EL1 and EL2 pointsare located, respectively, on the ξ axis at 388.810, 384.918,and 392.728 based on the length unit normalization LE-M

mentioned in Sect. 2.1. The temporarily captured segment ofthe translunar trajectory has fewer loops orbiting the Moonbut requires more energy than the cislunar one. Even so, thedeceleration from the temporary capture to the permanentcapture is smaller than for Hohmann transfer.

5 Conclusion

The low-energy cislunar and WSB trajectories are investi-gated in this paper from the viewpoint of the cislunar li-

bration point (LL1) and translunar libration point (LL2), re-spectively. According to the geometry of the instantaneousHill boundary, the equivalent LL1 point is defined as thecritical point connecting the two gravitational fields aroundthe Earth and Moon, while the equivalent LL2 point is de-fined as the critical point connecting interior and forbid-den regions. The locations of the equivalent equilibria andtheir Hamiltonian values are solved from the partial deriva-tive of the potential function with respect to the x compo-nent.

The systematical discussion on the Moon-captured en-ergy in the frame of a spatial analytical four-body model(i.e., SBCM) is implemented by numerical Poincaré map-ping, which is only focusing on the statistical features ofthe fuel cost and captured elements (like altitude of periap-


sis and eccentricity) rather than a specified Moon-capturedsegment.

The minimum-energy cislunar and translunar trajecto-ries are yielded by transiting the LL1 and LL2 points, re-spectively, in Sect. 3. The trajectories transiting the LL2

point are classified into the inner cislunar type essentiallypassing through the LL1 point, and the outer WSB typeconnecting the invariant manifolds near the EL1 (or EL2)point and unstable manifolds near the LL2 point. Moreover,it is demonstrated that the solar phasic angle β has pos-itive affects on the transfer opportunities: for the cislunarcase transiting the LL1 point, a whole Earth-to-Moon trans-fer trajectory can be achieved only within the β interval[77◦,109◦] ∪ [285◦,342◦]; for the outer WSB case transit-ing the LL2 point, a whole Earth-to-Moon transfer trajectorycan be achieved only within the β interval [21.8◦,23.3◦] ∪[201.5◦,203◦].

Compared with the only variable (i.e., β) to constructa transfer trajectory transiting the libration point, the haloorbit is employed to increase the transfer opportunities byintroducing another variable (i.e., series of points of haloorbit). Subsequently, a global investigation on the Earth-escaping and Moon-captured opportunities is implementedfor practical transfer trajectories transiting halo orbits nearthe LL1 and LL2 points, respectively. For the cislunar case ofthe transiting halo orbit near the LL1 point, a whole Earth-to-Moon transfer trajectory can be achieved only withinthe pairs (β, τ ) of ([95◦,150◦] ∪ [262◦,345◦]) × [0,1] and([100◦,200◦] ∪ [270◦,30◦]) × ([0.16,0.26] ∪ [0.47,0.58]);for the outer WSB case transiting the halo orbit near theLL2 point, a whole Earth-to-Moon transfer trajectory canbe achieved only within the pairs (β, τ ) of ([85◦,165◦] ∪[262◦,330◦]) × [0,1] and ([10◦,92◦] ∪ [192◦,268◦]) ×([0,0.26] ∪ [0.57,0.66]).

Acknowledgements The authors are very grateful to the anony-mous reviewer for helpful comments and suggestions on revising themanuscript. The research is supported by the National Natural Sci-ence Foundation of China (11172020), the National High Technol-ogy Research and Development Program of China (863 Program:2012AA120601), Talent Foundation supported by the FundamentalResearch Funds for the Central Universities, Aerospace Science andTechnology Innovation Foundation of China Aerospace Science Cor-poration, and Innovation Fund of China Academy of Space Technol-ogy.

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