Families of Cycler Trajectories

in the Earth-Moon System

Jordi Casoliva∗, Josep M. Mondelo†, Benjamin F. Villac‡,

Kenneth D. Mease§, Esther Barrabes¶ and Merce Olle‖

Motivated by the need for infrastructure in cis-Lunar space to support telecommunica-

tions, navigation, spacecraft servicing and astronaut rescue, periodic orbits in the Earth-

Moon system are computed, classified and assessed using relevant criteria. Using the planar

circular, restricted three-body model, cyclers – periodic orbits that pass near the Earth

as well as near or behind the Moon – are computed using three strategies. In the first, a

cycler is approximated by an elliptic Earth orbit, neglecting the lunar gravity, and then

differentially corrected to account for the lunar gravity. In the second strategy, a p-q res-

onant cycler is analytically approximated for small lunar mass as a seed for generating a

family of cyclers via differential correction and continuation. In the third strategy, cyclers

are constructed using homoclinic connections between unstable Lyapunov orbits associated

with a libration point. The utility of these cyclers is then assessed relative to the criteria

for cis-Lunar infrastructure.

I. Introduction

Space infrastructure beyond geostationary orbit will be needed to support Moon exploration and utiliza-tion and other future activities in the the Earth-Moon system. Anticipated elements of this infrastructure

are spacecraft in low maintenance orbits that regularly visit the vicinity of the Earth and the vicinity ofthe Moon. Such spacecraft would serve telecommunication, navigation or astronaut rescue purposes, forexamples. This need motivates the identification of appropriate periodic orbits in the Earth-Moon systemto serve as the backbone of such missions. The terms cycler and cycler trajectory will be used to refer to aperiodic orbit that passes near the Earth and near or behind the Moon.

Previous research1–5 has identified interesting cycler trajectories in the Earth-Moon system. Classesof periodic and resonant orbits have been determined via global search methods,1,6, 7 patched conic ap-proaches1,8, 9 and dynamical systems theory.10–20 The researches most directly related to our work is de-scribed further: The classification of planar, simple periodic symmetric families of orbits in mission designsin the Earth-Moon system was studied by Lo and Parker.16 This study considered the planar periodic orbitspierce orthogonally the x-axis in the rotating frame exactly twice per orbit. Continuation was used to exploreseveral families of this class of planar orbits in the restricted three-body problem, producing the invariantmanifolds of the unstable orbits in each of these families. While some of these orbits considered in this workcan be used for cyclers, we extend the scope of orbits types to higher energy cyclers and asymmetric orbits.

In Barrabes and Gomez,8,9 three-dimensional p-q resonant orbits close to periodic second species solutions(SSS) of the circular restricted three-body problem (CRTBP) were obtained. Analytical in- and out-mapsfor very small values of µ were computed for propagating initial conditions on a sphere of radius µα about

∗Ph.D Candidate, University of California, Irvine, CA 92697-3975, USA, Department of Mechanical and Aerospace Engi-neering, jcasoliv@uci.edu. Currently: Aerospace Engineer, Jet Propulsion Laboratory, Pasadena, CA 91109-8099, USA, Entry,Descent, and Landing / Aero Applications Group - Guidance, Navigation, and Control Section. Member AAS and AIAA.

†Associate Professor, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain, Departament de Matema-tiques, jmm@mat.uab.cat, Member AAS.

‡Assistant Professor, University of California, Irvine, CA 92697-3975, USA, Department of Mechanical and AerospaceEngineering, bvillac@uci.edu, Member AAS.

§Professor, University of California, Irvine, CA 92697-3975, USA, Department of Mechanical and Aerospace Engineering,kmease@uci.edu, Associate Fellow AIAA.

¶Assistant Professor, Universitat de Girona, 17071 Girona, Spain, Departament d’Informatica i Matematica Aplicada,barrabes@ima.udg.edu, Member AAS.

‖Associate Professor, Universitat Politecnica de Catalunya, 08028 Barcelona, Spain, Departament de Matematica AplicadaI, ETSEIB, merce.olle@upc.edu, Member AAS.

1 of 20

American Institute of Aeronautics and Astronautics

AIAA/AAS Astrodynamics Specialist Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6434

Copyright © 2008 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

the small primary, around a p-q resonant orbit and back to the same sphere. We use these analytic solutionsas seeds for computing cyclers for the Earth-Moon system.

More recently, an atlas of short-period and low-energy asymmetric/symmetric periodic orbits (PO) en-circling both the Earth and the Moon in the CRTBP was developed by Leiva and Briozzo.18 While thisatlas provides many examples of cycler trajectories, many of these orbits are seen to follow much the samepattern. In this paper, we propose to use the notion of homoclinic connections to some simple orbits torepresent, characterize and summarize whole classes of such short-lived families associated with librationpoint orbits.

The application of libration point dynamics has been very useful in the design of space missions such asNASA’s International Sun-Earth Explorer 3 (ISEE-3) in 1978, ESA’s SOlar and Heliospheric Observatory(SOHO) in 1995, Wilkinson Microwave Anisotropy Probe (WMAP) in 2001 and NASA’s Genesis12 in 2001.The L1 and L2 Lagrangian points allow to place permanent observatories of the Sun (like SOHO, aroundL1) or of the whole celestial sphere (like WMAP, around L2). Proofs of the existence of homoclinic orbits toLyapunov Periodic Orbits (LPO) around the L1 and L2 points for some particularly shaped homoclinic orbitshave been given.10,11 Most libration point missions launched up until the present14 consist of a nominaltrajectory and a transfer trajectory to it. The Genesis mission12 has been the first one to make use of aheteroclinic connection. The use of homoclinic and heteroclinic connections allow the design of complexmissions such as the Petit Grand Tour to the moons of Jupiter15 and low-energy transfers to the Moon.17

The computation of the homoclinic connections presented in this paper may be of use to similar missions.The objectives of this paper are to provide a methodology to compute periodic orbits that are candidate

cycler trajectories and assess them relative to criteria relevant to telecommunications, navigation and astro-naut rescue. Using the planar circular restricted three-body model, cyclers – periodic orbits that pass nearthe Earth as well as near or behind the Moon – are computed using three strategies. In the first, a cycleris approximated by an elliptic Earth orbit, neglecting the lunar gravity, and then differentially corrected toaccount for the lunar gravity. In the second strategy, a p-q resonant cycler is analytically approximated forsmall lunar mass as a seed for generating a family of cyclers via differential correction and continuation. Inthe third strategy, cyclers are constructed using homoclinic connections between unstable Lyapunov orbitsassociated with the L1 libration point. The results and methods are applicable to cycler trajectories in other3-body systems. This research complements previous work in linking theoretical classification of periodic or-bits to actual constraints present in the Earth-Moon system. This link provides a more structured approachthan current large scale numerical searches, and allows us to extract some guidelines for mission designers.Also, the investigation is not restrict to symmetric orbits, as in many studies.

Section II presents the background for the work. Section III defines the criteria used to classify the cyclertrajectories. Section IV describes generation of high energy near-Keplerian cyclers. Section V explains thegeneration of low energy cyclers. Finally, section VI is the summary and conclusions.

II. Background

In this section, we review the notations used for the dynamical model used, the planar, circular, restrictedthree-body problem, as well as the fundamental continuation methods used in the remainder.

II.A. Dynamical Models

In this paper, we considered the planar, circular, restricted three-body problem (PCR3BP) as a model ofthe dynamics of a spacecraft flying in the Earth-Moon system. While the Sun’s influence is important inthis system, the PCR3BP, provides a first approximation that contains all of the orbit classes of interest.One can then use differential correction methods to transfer sample trajectories to more realistic models, asshown in 7. Provided that the spacecraft trajectory does not extend too far beyond the vicinity of Earth,this model provides a first approximation to the actual trajectory that would be flown in the presence ofthe Sun and other gravitating bodies. The three-body problem is composed of a primary body, Earth, andsecondary body, the Moon, and a spacecraft. In the circular, restricted three-body model, the gravitationalforces of the primary and secondary bodies, but not the spacecraft, are accounted for, and the Earth andthe Moon are assumed to be in circular orbits about the Earth-Moon barycenter. Normally, the trajectoriesof a spacecraft are represented in the synodic frame whose origin is at the barycenter (i.e., center of mass)of the primary and secondary bodies and is rotating with an angular velocity such that the Earth and the

2 of 20

American Institute of Aeronautics and Astronautics

Moon are at fixed points on the x-axis. The coordinates of the Earth are (µ, 0) and those of the Moon are(µ-1, 0), where µ = 0.01215 is the Moon-Earth mass ratio.

The equations of motion for the restricted three-body are normalized such that the separation betweenthe primary and secondary bodies is one length unit and the orbit period is 2π time units and are given as

dX

dt= f(X) with X = [x y u v]

Tand f(X) =

[

u v 2v +∂Ω

∂x− 2u+

∂Ω

∂y

]T

(1)

where the potential function Ω is defined as

Ω =x2 + y2

2+

1 − µ

r1+µ

r2; r1 =

√

(x− µ)2 + y2; r2 =√

(x− µ+ 1)2 + y2 (2)

Finally, using Eqs. (2), the Jacobi constant is defined as

CJ(X) = 2 Ω − ||V||2 with V = (u v)T

(3)

where CJ is constant with respect to time for a given trajectory in the PCR3BP. Hence, a particle with aspecific Jacobi constant CJ in the PCR3BP cannot transfer to regions of space of different CJ (i.e., forbiddenregions) unless a non-conservative maneuver is performed.

For some computations, the Hamiltonian form of the PCR3BP will be used. In this formulation, momentaare defined by px = u− y, py = v + x, so the equations are dX/dt = g(X), with X = [ x y px py ],and

g(X) =

[

px + y py − x − ∂H

∂x− ∂H

∂y

]

with Hamiltonian

H(x, y, px, py) =1

2(p2

x + p2y) − xpy + ypx − 1 − µ

r1− µ

r2. (4)

In the following, the value of the Hamiltonian will be sometimes referred to as “energy”. The relationbetween the Jacobi constant and the Hamiltonian is

C = −2H. (5)

II.B. Stability

The state transition matrix, Φ, provides the derivatives of the state X(t) with respect to the initial stateX(ti) and satisfies the initial value problem:

Φ(t, ti) =∂f(X)

∂XΦ(t, ti) (6)

Φ(ti, ti) = I (7)

The stability of a periodic orbit may be determined by analyzing the eigenvalues of the state transition matrixafter a full period (i.e., the monodromy matrix). For the PCR3BP, the monodromy matrix is symplecticand, hence, its eigenvalues occur in reciprocal pairs. Moreover, due to the existence of the Jacobi constantCJ , two eigenvalues will be unity. Therefore, the eigenvalues of the monodromy matrix will be λ, 1/λ, 1, 1.The stability indices k used in this paper are defined as

k = λ+ 1/λ (8)

where k ∈ R. For stable orbits, the value of k is in the range |k| 6 2. For unstable orbits, |k| > 2 becausethe real component of at least one of the eigenvalues summed is greater than one.

3 of 20

American Institute of Aeronautics and Astronautics

II.C. Numerical Continuation

Families in the PCR3BP consist of a continuum of periodic orbits whose properties vary continuously fromone end of the family to the other. The Stromgren-Wintner’s natural termination principle states thepossibility of analytic continuation of any one-parameter family of periodic orbits in the restricted threebody problem as long as the solutions stay clear of a collision singularity and their periods remain bounded.All orbits in the same family may be uniquely identified by a single parameter of that family such as theperiod, Jacobi constant or one coordinate from the initial conditions. The fact that the families of periodicorbits are embedded in manifolds is due to the existence of the Jacobi constant and, hence, once a singleperiodic orbit is found, continuation methods may be used to traverse that family of the orbit.

The numerical continuation of families of periodic orbits (p.o.) is a well–known topic. It is usuallydone by applying an standard predictor–corrector or pseudo–arclenth method21 to a system of (nonlinear)equations that has the family of p.o. as solutions. As an example of two different approaches to constructsuch systems, see 22,23. The actual system of equations we have used for the continuation is:

H(x0) − h = 0,

g(x0) = 0,

φT (x0) − x0 = 0,

where H is the Hamiltonian of the PCR3BP, g(x) = 0 defines a Poincare section (in order to have a uniquepoint of the p.o. as solution of the equations), and φT (x) is the time–T flow of the PCR3BP. The unknownsare:

µ, h, T,Xo

where µ is the PCR3BP mass parameter, h is the energy level (value of the Hamiltonian) of the p.o., T is itsperiod and x0 = (x, y, px, py) is the vector of initial conditions (positions and momenta). Thus there are 6equations with 7 unknowns. By fixing the value of one of the unknowns µ, h, T , a family of p.o. with eitherµ, h or T constant can be followed.

III. Criteria for Cyclers

In this section, criteria for assessing the utility of a cycler are presented.

The criteria are:

1. Apogee radius (i.e., maximum distance relative to the Earth)

2. Perigee radius (i.e., minimum distance relative to the Earth)

3. Periselene radius (i.e., minimum distance from the Moon)

4. Velocities at both perigee and periselene radii

5. Period of the cycler

6. Stability index k

7. Energy (i.e., Jacobi constant)

8. Backside of the Moon coverage.

The apogee radius (i.e., criterion 1) bounds the size of the cycler trajectory with respect to the Earth. Criteria2, 3 and 4 are important to quantify the cycler insertion costs and can be transformed into the propellantrequired for transferring mass to and from the cycler trajectories. Criterion 6 indicates how sensitive thecycler trajectory is to perturbations and, thus, it is desired that k 6 2. In general, station-keeping costsare smaller if the cycler trajectory is stable. Criterion 7 is important in case a particular mission required atransfer from one cycler trajectory to another (or a backup cycler). Therefore, it would be desirable to choosetwo cyclers with similar Jacobi constant CJ to minimize the fuel burn required for the transfer. Criterion 8could be relevant for communications.

4 of 20

American Institute of Aeronautics and Astronautics

IV. Second Species Cyclers

This section discusses the computation and main characteristics of cycler trajectories solutions which aregenerated from elliptical Earth orbits when the Moon is ignored, as well as patched conic approximations.Attention is given to orbits satisfying a resonant relation such that orbits are then periodic in the inertialframe, and thus provide an advantage with respect to navigation or the design of constellations.

First, a review of the constraints on the resonance relations and a numerical approximation of resonantcyclers are presented. Then, a more theoretically rooted method8,9 starting from an analytical estimate isdiscussed. As opposed to most previous studies classifying periodic orbits, the solutions described in thissection take into account the restrictions implied by their application to spaceflight.

IV.A. Cycler Constraints on Resonances

Ignoring the influence of the Moon’s gravitational pull on the spacecraft, while still investigating the motionof a spacecraft in the synodic frame, one can define a resonance relation between the spacecraft motion andthe rotational motion of the frame. We say that the spacecraft is in a p-q resonant orbit when

p TM = q Ts/c (9)

where p and q are relative prime integers, TM and Ts/c represent the period of the Moon and the spacecraftwhen defined in a sidereala frame. In effect, such resonance relations imply that the spacecraft will completeq times its (inertial) elliptic orbit while the Moon will have completed p revolutions around the Earth (againin inertial space, i.e., p×28 days in physical units). A resonant trajectory will thus be both periodic inthe synodic and sidereal frames. The periodicity in the sidereal frame has to be understood as a repeatingrelative configuration of the Earth-Moon-spacecraft system with respect to this frame. In the rotating frame,the p-q resonance relation implies a periodic orbit with period q Ts/c.

Given Kepler’s third law relating period to semi-major axis, the resonance relation implies a constrainton the size of the spacecraft orbit (or equivalently its “Keplerian” energy). More precisely, the semi-majoraxis and energy of the spacecraft must satisfy

as/c =

(

p

q

)2/3

and Es/c =µE

4

(

q

p

)2/3

(10)

where the semi-major axis of the Moon is one in the unit system selected. Table 1 shows the resonantsemi-major axis for several values of p and q.

Table 1. Spacecraft semi-major axis (relative to Earth) in normalized units for low order reso-nance. The forbidden resonances (due to the cycler constraint) are shaded in grey. The lowest order

resonances are underlined†.

p-q 1 2 3 4 5 6 7 8 9

1 1.000 0.630 0.481

2 1.587 1.000 0.763 0.630 0.543 0.481

3 2.080 1.310 1.000 0.825 0.711 0.630 0.568 0.520 0.481

4 2.520 1.587 1.211 1.000 0.862 0.763 0.689 0.630 0.582

5 2.924 1.842 1.406 1.160 1.000 0.886 0.800 0.731 0.676

† The value of semi-major axis for non-relative prime integers has been given for ease of reference. Howeverin that case the resonance value should be considered to be given by the “reduced fraction”. For examplea 6-3 resonance is really a 2-1 resonance.

Note in particular, that when p < q, the spacecraft semi-major axis is less than 1 and it is possiblethat the spacecraft does not encircle the Earth and Moon within its orbit. More precisely, a spacecraft willencircle the Moon – and the trajectory will thus be a candidate cycler – if 2 as/c > 1 + ǫ, where epsilon

aThat is, inertial space. Note that, implicitly, we assume the spacecraft on an elliptic orbit around the Earth.

5 of 20

American Institute of Aeronautics and Astronautics

represents the minimum allowed distances of the spacecraft from the center of the Earth and the Moon.Taking ǫ ≃ 0.02289, that is, allowing ∼300 km altitude at closest approach from the Earth and ∼70 kmaround the Moon, the physical constraint on the resonances is then

q < p

(

2

1 + ǫ

)3/2

(11)

that is, q 2.8284 p. This constraint is represented graphically in Figure 1. The resonances are representedas a regular lattice of dots in (p-q)-space. The allowable resonances are those outside the shaded area. The“forbidden resonances” will thus be discarded in the remainder.

Figure 1. Graphical representation of resonance restrictions.

For the allowed resonances (i.e allowing the spacecraft trajectory to encircle both the Earth and Moon),a fixed semi-major axis implies constraints on the maximum apoapsis of the trajectory. Specifically, theapogee ra and perigee rp radii satisfy

ra = 2

(

p

q

)2/3

− rp with ra ≤(

p

q

)2/3

− RE (12)

where RE represents an effective radius of the Earth (Earth radius plus atmosphere or security boundary,which is considered to be on the order of 7000 km). However, note that the minimum periselene constraintcan be controlled with phasing and should be distinguished from the above constraints which pertain to theEarth. In particular, note that the set of resonant orbits, as defined above, is only parametrized by theresonance relation, the eccentricity e and longitude of the perigee ω (since we only consider planar motion).

While the resonance relation in Eq. (9) is a constraint on the energy of a trajectory, the constraintdefined in Eq. (12) can be understood as an angular momentum constraint, which is more easily expressedin terms of eccentricity

RM

2

(

p

q

)2/3

< e < 1 − RE

2

(

p

q

)2/3

with h =√

µ a (1 − e2) (13)

Note in particular that the minimum altitude allowed at the Moon (included in the term RM ) controlsthe minimum eccentricity of the orbit (or maximum allowed angular momentum), while the safety marginaround the Earth controls the maximum eccentricity (minimum angular momentum). The resonances nearthe boundary line q = p 23/2 imply close fly-bys with the Earth and Moon and may thus not be presentwhen the influence of the Moon is considered.

6 of 20

American Institute of Aeronautics and Astronautics

IV.B. Cycler via Differential Correction and Continuation from 2-Body Orbits

One approach to generating a first approximation for a resonant cycler is to ignore the Moon’s influence on thespacecraft trajectory, so that the spacecraft follows an elliptical orbit around the Earth. This approximationprovides initial conditions for resonant orbits which can then be numerically differentially corrected towardperiodic orbits. Figure 2 presents a sample trajectory obtained with this approach for three resonances 1-2,2-3 and 3-4.

(a) 1-2 resonance (b) 2-3 resonance (c) 3-4 resonance

(d) 1-2 resonance (corrected) (e) 2-3 resonance (corrected) (f) 3-4 resonance (corrected)

Figure 2. Top Row: sample initial conditions from a two-body approximation for (a) 1-2 (b) 2-3 and (c) 3-4 resonances.Bottom Row: corresponding numerically differentially corrected periodic orbits.

Assuming these orbits to be truly periodic in the restricted problem, they belong to a family of periodicorbits which can generically be parametrized by the Jacobi constant CJ . One can then numerically continuesuch orbits as a function of CJ . This is illustrated in Fig. 3. In particular, differential continuation allowsone to increase the initial perigee, while decreasing the apogee toward smaller fly-by altitude at the Moon.This affects the period of the orbit and one looses the initial resonance relation. However, as the familyevolves, other resonances are encountered, albeit higher order resonances. For example, the family generatedfrom the 2 : 3 resonance encounters the 22 : 18, the 18 : 15 and 26 : 21 resonances. These higher orderresonances can be understood as lower order resonances plus a shift along the orbit. In the case of the 18 : 15resonance is indeed a 9 : 5 resonance plus a shift of half of the period for the cycler. For the non-resonantcyclers, the motion in inertial space is quasi-periodic and result in a secular drift of the orbit relative to afixed direction.

Note that while the above approach is sufficient for cyclers with p > q, it may not provide a sufficientlygood approximation for resonances with smaller semi-major axis (p < q). This is especially true for resonanceswith small semi-major axis such as the 3 : 7 and 3 : 8 resonances, for which the apogee is near the Moon.From our experience, these approximation fails when the semi-major axis is on the order of 0.6 or below.This approach does not allow also a fine control on the longitude of perigee as this quantity generally driftsduring the correction process.

7 of 20

American Institute of Aeronautics and Astronautics

(a) 1-2 (b) 2-3 (c) 3-4

Figure 3. Sample periodic orbits along the continuation of the 2-3 resonance provided by the two-body approximationmethod.

Alternatively, one can start from the two-body approximation and continue the solution as a function ofthe mass parameter, µ. This approach leads to better success when the semi-major axis is still sufficientlylarge and allows for a better control on the longitude of the ascending node. However, this approach failsfor limiting resonances, such as the 3 : 8 resonances for which the apogee must lie very close to the Moon.

Of the resonance computed, the resonances with p > q present large apogee distance beyond the Moonorbit and may be less interesting for applications. Given their higher energies, the transfer to these cyclers ismore expensive and the Sun’s influence should be analyzed further on these trajectories. Of the lower orderresonance with p < q, the 2 : 3 and 3 : 4 cyclers provide a good coverage of the Earth-Moon system, withoutinvolving too large apoapsis.

IV.C. Cyclers via Differential Correction and Continuation from a Patched Conic Orbits

In order to improve over the two-body approximation, we considered the continuation of patched-conicapproximation to second species solutions. Trajectories that encircle both the Earth and Moon are partof families which pass through the Moon and are known as second species solution in the terminology ofPoincare.24 Second species solutions are thus periodic orbits generated from segments of two-body orbits asthe mass parameter µ is let to tend to zero. Henon24 provides a systematic classification of such an arc thatcan be continued to non-zero mass ratio. Except for the physical restriction encountered while increasingthe µ value to that of the Earth-Moon system, these orbits are the cyclers we are looking for which donot have the secular drift present in the direct approach mentioned above. While Henon’s work24 providesa classification in terms of generating arcs, for actual continuation, one needs numerical values of initialconditions to generate these periodic orbit families. For these, we use the work of Barrabes and Gomez.8,9

IV.C.1. Approximate Initial Conditions

In Refs. 8,9, analytical approximation based on a patched conic idea were used to generate three-dimensionalp-q resonant orbits close to periodic second species solutions of the restricted three-body problem. Analyticalin- and out-maps for very small values of µ were computed for propagating initial conditions on a sphere ofradius µα about the small primary, around a p-q resonant orbit and back to the same sphere. The out-mapdoes forward time propagation and the in-map, backward time propagation. Analytical expressions in powersof µ are given8,9 for both mappings as well as the matching conditions. These analytical expressions yieldinitial conditions of orbits that will be “periodic” with an error of the order µ1−α, for some α ∈ (1/3, 1/2).Since µ → 0, the inner solution and the outer solution will collide with the small primary and, thus, theseorbits will be close to the second species solutions.

Let Xi = (xi yi ui vi)T be the initial conditions of a planar p-q resonant orbit in a synodical reference

system, where the secondary body M is located at rM = (µ − 1 0)T . The spacecraft P leaves a circle C of

8 of 20

American Institute of Aeronautics and Astronautics

radius µα around the small primary M. Thus, the initial conditions can be written as8,9

xi = µ− 1 + µα cos θ yi = µα sin θ (14)

ui = v cosψ vi = v sinψ

Barrabes and Gomez8,9 obtained conditions on θ, ψ, v and the corresponding Jacobi constant CJ whichensure that the orbit is a p-q resonant orbit close to a p.o. by matching the analytical expressions of the in-and out-maps up to order µα. For each p, q ∈ N with p/q 6 2

√2, there is a family of p-q resonant orbits

with the Jacobi constant, CJ , within the interval

(

p

q

)2

3

− 2

√

2 −(

p

q

)2

3

< CJ <

(

p

q

)2

3

+ 2

√

2 −(

p

q

)2

3

(15)

For each value of CJ within the above set, there are two admissible values of the polar angle of the velocityψ given by the relation

sinψ =2 − CJ +

(

pq

)2

3

2√

3 − CJ

(16)

Therefore, for every admissible pair (CJ , ψ), there exists one value of θ such that the initial conditions inEq. (14) satisfy the corresponding matching conditions between the in- and out-maps. Hence, θ must satisfythe nonlinear constraint below

2√3 − CJ

(

cos θ sin2(θ − ψ) − cosψ cos(θ − ψ))

+ sin3(θ − ψ) = 0 (17)

such that cos(θ − ψ) > 0, i.e., the spacecraft is moving away from circle C. Finally, the total time requiredto come back close to the initial condition defined in Eq. (14) is

T = 2πq + O(µα) (18)

IV.C.2. Numerical Results and Discussion

The approximate initial conditions for p-q resonant orbits discussed in the previous paragraph and defined byEqs. (14-18)are used as initial guesses. While the initial condition provided by the two-body approximationcan be continued21 to the restricted problem on a dense set,24 only finitely many CJ values provide actualperiodic motion in the case of p-q resonant orbits, as defined above. This is due to the close fly-by of theMoon imposed in the definition of such orbits. Since these approximate initial conditions are only valid forvery small values of µ and do not yield perfectly periodic orbits, the initial conditions had to be differentiallycorrected using a grid of CJ values. Only a few initial conditions per p-q pair at µ = 10−6 yield periodicorbits. Differentially correcting such orbits presented several challenges. First, the small value of the massparameter requires significant accuracy in the integration and very small step in the continuation parameter.

Table 2 shows the cycler designation, p-q values, Jacobi constant CJ , period T and initial conditions xi

and vi, the application (communication/navigation or transportation) of the cycler and its stability indexk for several cycler trajectories. Figure 4 shows the plots of the cycler trajectories from Table 2. Thedesignation of the cycler trajectories obtained via p-q resonant orbits is composed of 2 digits (the p-q values)followed by a letter (from “a” to “z”). For given p-q values, there are only a discrete set of p-q resonant orbitsthat will be periodic and, thus, the letter “a” would represent the cycler trajectory within a p-q family thathas the lowest Jacobi constant value. The orbits resulting from these initial conditions is shown in Figure 4.

Then, to look for resonant periodic orbits in the Earth-Moon system, continuation21 in the mass pa-rameter µ was performed at fixed T . While this continuation is the most natural to preserve the resonantcharacter of these orbits, it leads in most cases to drive the periodic orbits to the singularity at the centerof the Moon, showing that the initial resonance relation on these particular families is in fact not preserved.However, such resonance relation may be present in families with different phasing and presenting moredistant fly-bys. For example, the cycler shown on Fig. 4(f) cannot be continued to the mass parameter ofthe value of the Earth-Moon system, even though it is very similar to the 2 : 3 resonance presented in Fig.2. These cyclers are in fact different only by their longitude of perigee.

9 of 20

American Institute of Aeronautics and Astronautics

Table 2. Cycler designation, p-q values, Jacobi constant¯oldsymbolCJ , period and initial conditions, application and

stability index for several cycler trajectories

Designation p-q CJ and T Xi* Vi

† k‡

12a 1-2-0.4048949508278787 -0.9997842236429277 -1.0475686168407430

994.821412.5729004699819580 0.0000000000000000 1.5221048236500363

21a 2-10.3044238301466371 -0.9994542367695188 0.0000000000159674

2.02146.2807379868905375 0.0000000000000000 1.6429377286480178

23a 2-3-1.4624706218555543 -0.9997040086087932 -0.0000000000029493

1.840518.8645742008117736 0.0000000000000000 2.1140593042383320

23b 2-3-0.3856270265962789 -0.9953809710199844 -0.9554688344071062

-0.775318.8497995179817757 0.0000000000000000 1.5726409617865593

32a 3-2-0.3403221450450835 -0.9999035023356472 0.0000000000131068

6.381812.5363376944500722 0.0000000000000000 1.8333742370247279

32b 3-22.0635340336761394 -1.0188148478462549 -0.7183851038146349

1.568112.5660196280208911 0.0000000000000000 0.6492612209948345

52a 5-21.0461882704974470 -0.9994423797258251 0.0000000000459148

2.036612.5651492405106922 0.0000000000000000 1.3990717541095201

54a 5-4-0.5902501452788234 -0.9988849982363450 -0.2709574260065351

-1.319225.1304852528305673 0.0000000000000000 1.8758004354491455

54b 5-4-0.6598717597930506 -0.9905419593393083 -0.1198961389475177

1.994325.1321450585584110 0.0000000000000000 1.9094434196183308

73a 7-30.8957501590757784 -0.9954265899784440 -0.2486030886355384

1.879918.8492803402344329 0.0000000000000000 1.4293154529931373

* Initial position in the x and y coordinates.† Initial velocity in the x and y coordinates.‡ Stability index k as defined in Eq. (8).

Another example is presented in Figs. 5 and 6, where the orbit represented in blue has been generatedwith the above patched conic approximation while the red orbit correspond to its continuation at fixed periodas a function of T . The longitude of the perigee is shifter by 90o and the “cycler” looses actually its propertyof fly-by the backside of the Moon.

As with the 2-body approximation, one can also continue these families in CJ so as to increase theperiselene distance. Then for a family member sufficiently far from the Moon, we can continue the familyat fixed CJ until the mass ratio matches the value for the Moon. The resulting periodic orbit can thenbe continued at fixed µ by varying CJ and investigate the various higher order resonance relations present.As opposed to the pure classification of second species solutions, not all the allowable resonance relationindicated by the two-body approximation are allowed for cyclers in the Earth-Moon systems. There is aclose relation between the phasing of these trajectories and the existing resonance that future work willfurther clarifies.

10 of 20

American Institute of Aeronautics and Astronautics

−3 −2 −1 0 1 2

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

X−coordinate

Y−

co

ord

ina

te

(a) Designation 12a

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

X−coordinate

Y−

coord

inate

(b) Designation 21a

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

X−coordinate

Y−

co

ord

ina

te

(c) Designation 23a

−1.5 −1 −0.5 0 0.5 1 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

X−coordinate

Y−

co

ord

ina

te

(d) Designation 23b

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

X−coordinate

Y−

coord

inate

(e) Designation 32a

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1

X−coordinate

Y−

co

ord

ina

te

(f) Designation 32b

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

X−coordinate

Y−

co

ord

ina

te

(g) Designation 52a

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X−coordinate

Y−

coord

inate

(h) Designation 54a

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X−coordinate

Y−

co

ord

ina

te

(i) Designation 73a

Figure 4. Periodic orbits computed for different p-q resonances with µ = 1e−6. The gray and blue circles depict theMoon and Earth respectively.

V. Homoclinic Type Cyclers

In this section we explore a different class of cycler trajectories which may be more suited for cargotransportation applications or servicing missions to the libration points. Notably, we proposed to use thecontinuation of homoclinic connections to unstable periodic orbit of the third species,24 as a mean to organizethe infinite number of periodic orbits that exist in their vicinity; some of which having been considered in.18

This is illustrated in the case of the Lyapunov family of periodic orbits originating from the L1 point. Notethat the energy value in the following refers to the Hamiltonian, that is 0.5CJ .

As it has been mentioned in the introduction, theoretical results concerning homoclinic connectionssimilar to the ones computed here can be found in.10,11 Numerical computations can be found in 25. Other

11 of 20

American Institute of Aeronautics and Astronautics

−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

X−coordinate

Y−

coord

inate

Figure 5. Argument of perigee shift with continuation in the mass parameter µ and fixed period T .

2 4 6 8 10 12

x 10−3

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

Mass parameter µ

Init

ial

po

siti

on

xi

2 4 6 8 10 12

x 10−3

−10

−8

−6

−4

−2

0x 10

−15

Mass parameter µ

Init

ial

posi

tion y

i

2 4 6 8 10 12

x 10−3

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0x 10

−8

Mass parameter µ

Init

ial

vel

oci

ty u

i

2 4 6 8 10 12

x 10−3

1.8

2

2.2

2.4

2.6

2.8

Mass parameter µ

Init

ial

vel

oci

ty v

i

2 4 6 8 10 12

x 10−3

5.5

6

6.5

7

Mass parameter µ

Per

iod T

2 4 6 8 10 12

x 10−3

3.4

3.5

3.6

3.7

3.8

3.9

Mass parameter µ

Sta

bil

ity i

ndex

k

Figure 6. Continuation in the mass parameter µ and fixed period T .

applications of homoclinic and heteroclinic connections of p.o. can be found in 13.

V.A. Homoclinic connections continuation

The invariant stable manifold of an object is mathematically defined as the set of trajectories that asymp-totically approach the object forward in time. They get close to the base object at an exponential rate.The invariant unstable manifold of an object is defined in the same way but backward in time. It thuscontains trajectories that depart from the base object at an exponential rate. If the object is a nominal

12 of 20

American Institute of Aeronautics and Astronautics

orbit, invariant (stable and unstable) manifolds provide natural conduits from which we can choose suitabletransfer and departing trajectories. When the base object is a periodic orbit, its invariant manifolds looklike two-dimensional tubes. Fig. 7 left shows the stable and unstable manifolds of a planar Lyapunov orbitassociated with the L1 libration point in the x− y-plane.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

Moon Earth

Ws

Wu

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

py

y

He1 He2He3

He4

Wu

Ws

Figure 7. Left: tubes corresponding to branches in x > 0 of the unstable (W u, red) and stable (W s, blue) manifoldsof the planar Lyapunov p.o. of energy −1.5921. Right: y, py projection of the 4th intersection of W u (blue curve) andthe 3rd intersection of W s (red curve) with the Poincare section x = 0.

A homoclinic connection of an object is a trajectory that asymptotically approaches the object bothforward and backward in time. Homoclinic connections can be computed numerically by matching thestable and unstable manifold tubes in a surface of section as follows. In Fig. 7 are represented the branchesin x > 0 of the unstable (Wu) and stable (W s) manifold tubes of the L1 Lyapunov orbit of energy −1.5921.When we compute the 4th intersection of Wu with the Poincare section Σ = x = 0 and the 3rd intersectionof W s with the same section, we obtain the right plot of Fig. 7. In this plot, we can find find 4 points ofintersection between Wu and W s. All these points asymptotically go to the Lyapunov p.o. when propagatedboth backward and forward in time. They therefore correspond to four homoclinic connections of theLyapunov p.o., which we will denote as Hei, i = 1, 2, 3, 4. These four homoclinic connections are shown inFig. 8. Note that He1 and He3 are self–symmetric with respect to y = 0, whereas He2 and He4 are symmetricto each other with respect to this same axis.

Due to the Hamiltonian character of the PCR3BP, periodic orbits are not isolated but embedded in one–parametric families. The same happens with their homoclinic connections. The Hei connections of Fig. 8are thus embedded in families, which we will denote again by Hei, i = 1, 2, 3, 4. We have followed thesefamilies using the methodology developed in 26. This methodology essentially consists of continuation witha standard predictor–corrector algorithm21 on the following system of (nonlinear) equations:

H(x) − h = 0,

g1(x) = 0,

φT (x) − x = 0,

‖vu‖2 − 1 = 0,

DφT (x)vu − Λuvu = 0,

‖vs‖2 − 1 = 0,

DφT (x)vs − Λsvs = 0,

g2

(

φT u

(

ψu(θu, ξ0)

)

)

= 0,

g2

(

φT s

(

ψs(θs, ξ0)

)

)

= 0,

φT u

(

ψu(θu, ξ0)

)

− φT s

(

ψs(θs, ξ0)

)

= 0,

whereH is the Hamiltonian of the PCR3BP, x = (x, y, px, py) is a vector of positions and momenta, g1(x) = 0defines a Poincare section for the p.o., φT denotes the time–T flow of the PCR3BP, g2(x) = 0 defines a

13 of 20

American Institute of Aeronautics and Astronautics

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

He1 connection

Moon Earth

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

He2 connection

Moon Earth

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

He3 connection

Moon Earth

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

He4 connection

Moon Earth

Figure 8. Homoclinic connections Hei, i = 1, 2, 3, 4, of energy −1.5921.

Poincare section for the connection, ξ0 is a scalar quantity which is kept fixed to a small value (e.g. 10−6),ψ

u(θ, ξ) denotes a parametrization of the unstable manifold tube, with θ a phase and ξ a scalar, and ψ

s(θ, ξ)

denotes a parametrization of the stable manifold tube. The unknowns are

h, T,x,Λu,vu,Λs,vs, θu, Tu, θs, T s,

being h the energy (value of the Hamiltonian), T the period of the p.o., x the initial conditions of thep.o. (positions and momenta), Λu,vu the unstable eigenvalue and eigenvector of the monodromy matrixDφ(x), Λs,vs the stable ones, Tu the time necessary to reach the section g2(x) = 0 from the linearapproximation of the unstable manifold, and T s the same for the stable one.

V.B. Results and discussion

Some connections obtained by this continuation procedure along the He1 family are shown in Fig. 9. TheHei families of connections for i = 2, 3, 4 evolve in a similar way. It can be observed that, as energy increases,the periodic (Lyapunov) orbit approaches the Moon, and at the same time the connection approaches theEarth.

This makes these connections interesting as cycler trajectories. An spacecraft can be placed in theLyapunov orbit, where it will be regularly flying by the Moon. When desired, the homoclinic connectioncan be taken, and this will give five Earth flyby opportunities before going back to the Lyapunov p.o. Asopposed to the previous class of cycler trajectories, this strategy allows for multiple encounter of the Moonbefore flying-by the Earth.

Of course, the Lyapunov p.o. needs station-keeping since it is unstable. Moreover, in order to take theconnection, a small maneuver is necessary in order to insert the spacecraft with the right phase on the

14 of 20

American Institute of Aeronautics and Astronautics

unstable manifold. Estimating the maneuver ∆V is a task for future work. A similar strategy could also beused to transfer the spacecraft onto Moon-bound orbits, such as the prograde orbit generated from the lowaltitude circular orbits around the Moon??.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

energy -1.5653

Moon Earth

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

energy -1.5511

Moon Earth

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

energy -1.5305

Moon Earth

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

energy -1.5005

Moon Earth

Figure 9. Evolution of the He1 family of homoclinic connections with respect to energy.

Of all the computed connections of the He1 family, we display in Fig. 10 the one with the closestapproaches to Earth and Moon. The p.o. part (resp. the connection part) is represented in blue (resp. in red).Table 3 displays some parameters associated to the pericenters and apocenters of this connection. Amongthese parameters there are some orbital elements, which are given with respect to an inertial frame centeredeither in the Earth or the Moon. The numbering of pericenters and apocenters corresponds to that givenin Fig. 10, and we will use this same numbering for other connections of the family. For the connection ofFig. 10, the period of the Lyapunov orbit is 29.1640 days. The time needed in order to follow the connectionpart from periselene 1 to periselene 19 is 113.6319 days. We will refer to this quantity as the connectionflight time from now on.

It can be seen that, at periselene 1, the Lyapunov orbit flies by the Moon at a distance of 6325 km. Whenviewed in the 2–body problem Moon–s/c, this periselene turns out to be the aposelene of an elliptical orbitwith a semi-major axis of 3320 km and eccentricity 0.905. This because the velocity in the inertial frameis far smaller that in the rotating frame, since the Coriolis correction at this point is roughly the orbitalvelocity of the Moon, which is opposite the direction of movement. A similar phenomenon exists in Hill’sproblem 27. Note that, although in the Moon–s/c 2–body problem the orbit of this point would never reachthe Moon’s sphere of influence (which is roughly 67000 km), in the PCR3BP it goes far beyond.

For the connection part, it is seen in Table 3 that the perigee and apogee distances are roughly equal.When viewed in the two-body Earth–spacecraft system, the minimum perigees (numbers 8 and 12) correspondto the perigees of elliptical orbits of semi-major axis 201832 km and eccentricity 0.66. In order to rendezvouswith a spacecraft in this perigee from a circular orbit around the Earth of radius 67808 km, a ∆v of 703m/s

15 of 20

American Institute of Aeronautics and Astronautics

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

Moon Earth 1

2

3

4

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

y

x

Moon Earth 1,19

2

3

4

5 6

7

8

9

10

11

12

13

14 15

16

17

18

Figure 10. Pericenters and apocenters of the homoclinic connection of energy −1.4502. Left: p.o. part. Right: connectionpart.

would be required.We end this section by showing the evolution of several parameters along the He1 family that could be

of interest in the selection of a particular trajectory for specific purposes.In Fig. 11, the evolution of r, v, e at periselene 1 of the p.o. (closest approach to the Moon) is shown. In

order to explain why the velocity decreases to zero and then increases again, note that the motion in thePCR3BP (rotating frame) is always retrograde at periselene 1. For low energies, the PCR3BP velocity is nothigh enough to compensate for the rotation of the axes, so the inertial motion is direct. As energy increases,the PCR3BP velocity also increases, and the inertial velocity decreases until it reaches the zero value atenergy −1.473. This energy also gives a maximum in eccentricity (Fig. 11 right). From this energy on, themotion is retrograde and the inertial velocity increases. In Fig. 11 right it can also be observed that, inthe two-body Moon–spacecraft, periselene 1 corresponds to an hyperbolic trajectory for low energies, whichbecomes elliptic as v decreases, goes through circular, becomes a collision trajectory when v = 0 (changefrom direct to retrograde), and finally becomes elliptic again.

0

0.1

0.2

0.3

0.4

0.5

0.6

-1.56 -1.54 -1.52 -1.5 -1.48 -1.46 -1.44 5000

10000

15000

20000

25000

30000

35000

40000

vp

eri p

.o. (k

m/s

)

r pe

ri p

.o. (k

m)

energy

vperi p.o.rperi p.o.

5000

10000

15000

20000

25000

30000

35000

40000

-1.56 -1.54 -1.52 -1.5 -1.48 -1.46 -1.44 0

0.2

0.4

0.6

0.8

1

1.2

r pe

ri p

.o. (k

m)

ep

eri p

.o.

energy

rperi p.o.eperi p.o.

Figure 11. For periselene 1 of the p.o., evolution of r, v (left) and r, e (right) with respect to energy along the He1family.

Fig. 12 gives similar plots for perigee 8 of the connection part (closest approach to Earth). In this case,since we are near Earth, the velocity of the rotating frame is smaller and it does not affect the orbitalelements as when we are close to the Moon. It can be observed that perigee radius decreases as perigeevelocity increases. The eccentricity increases with energy, ranging from 0.476 to 0.664, and has not been

16 of 20

American Institute of Aeronautics and Astronautics

displayed. Instead, the right plot of Fig. 12 displays, as an estimation of the rendezvous cost from Earth,the ∆v of perigee 8 with respect to a circular trajectory of the same radius.

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

-1.56 -1.54 -1.52 -1.5 -1.48 -1.46 -1.44 65000

70000

75000

80000

85000

90000

95000

100000

105000

110000

vp

eri c

on

. (k

m/s

)

r pe

ri c

on

. (k

m)

energy

vperi con.rperi con.

65000

70000

75000

80000

85000

90000

95000

100000

105000

110000

-1.56-1.54-1.52 -1.5 -1.48-1.46-1.44 0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

r pe

ri c

on

. (k

m)

∆v

pe

ri c

on

. (k

m/s

)

energy

rperi con.∆vperi con.

Figure 12. For perigee 8 of the connection, evolution of r, v (left) and r, ∆V (right) with respect to energy along theHe1 family.

Fig. 13 displays the evolution of the period of the p.o. and the connection flight time as a function ofenergy. Recall that the connection flight time is computed as the time from periselene 1 to periselene 19 (seeFig. 10 right). It can be observed that the last computed connection represented in Fig. 10 has both thehighest period and the highest connection flight time. These times can be lowered at the expense of raisingthe perigee and periselene altitudes.

12

14

16

18

20

22

24

26

28

30

-1.56 -1.54 -1.52 -1.5 -1.48 -1.46 -1.44

T (

days)

energy

o.p.

80

85

90

95

100

105

110

115

-1.56 -1.54 -1.52 -1.5 -1.48 -1.46 -1.44

Tco (

days)

energy

connection

Figure 13. Evolution of the period of the o.p. and the connection flight time with respect to energy along the He1family.

17 of 20

American Institute of Aeronautics and Astronautics

p.o. of energy −1.450162

lbl Tflight r$ v$ a$ e$ ω$

1 0.000 6325.459 0.271 3320.262 0.90511 −0.000

2 8.983 259919.372 0.952 −5649.305 2.11119 70.184

3 14.582 138300.201 1.466 −2359.010 59.62637 0.000

4 20.181 259919.372 0.952 −5649.305 2.11119 −70.184

connection of energy −1.450162

lbl bdy Tflight rbdy vbdy abdy ebdy ωbdy

1 $ 0.000 6573.556 0.249 3429.393 0.91708 -0.760

2 $ 8.565 245836.046 0.952 -5658.111 1.00021 70.381

3 $ 13.730 142540.921 1.476 -2323.020 62.31635 3.139

4 $ 19.935 300136.002 0.940 -5763.960 6.04401 -67.513

5 $ 29.939 49872.612 0.820 -10300.291 1.07234 -54.055

6 34.250 70971.632 3.063 215364.185 0.67046 -62.458

7 39.973 357925.556 0.609 214728.164 0.66695 41.848

8 45.666 67808.075 3.128 201832.084 0.66404 147.477

9 51.236 353798.282 0.602 210749.263 0.67892 -106.097

10 56.816 68883.803 3.122 218603.529 0.68489 -0.000

11 62.396 353798.284 0.602 210749.266 0.67892 106.097

12 67.965 67808.081 3.128 201832.090 0.66404 -147.477

13 73.659 357925.559 0.609 214728.168 0.66695 -41.848

14 79.382 70971.637 3.063 215364.190 0.67046 62.458

15 $ 83.693 49872.615 0.820 -10300.290 1.07234 54.055

16 $ 93.697 300136.005 0.940 -5763.960 6.04401 67.513

17 $ 99.902 142540.921 1.476 -2323.020 62.31635 -3.139

18 $ 105.067 245836.045 0.952 -5658.111 1.00021 -70.381

19 $ 113.632 6573.556 0.249 3429.393 0.91708 0.760

Table 3. Flight times and orbital elements of the pericenters and apocenters of the homoclinic connection of energy−1.4502. Flight times are given in days and measured from periselene 1, both for the p.o. and connection parts. Theunits of r, a are km, and v is in km/s. The argument of pericenter ω is given in degrees and measured counterclockwisefrom the positive x direction.

18 of 20

American Institute of Aeronautics and Astronautics

VI. Summary and Conclusions

A methodology to classify large classes of cycler trajectories in the Earth-Moon systems has been pre-sented as well as a discussion of their potential applications. It has been shown that while certain resonantcyclers were allowed from a 2-body problem viewpoint, their existence in the Earth-Moon system is not guar-anteed. As a rule of thumb, a resonant ellipse with semi-major axis less that 0.6 the Earth-Moon distancemay not exist in the Earth-Moon system. This includes the 2 : 5, 3 : 7 or 3 : 8 resonances.

On the other hand, many cycler trajectories closely shadow two-body motion, which provide a usefulbackbone for preliminary mission analysis. These second species cyclers provide a global, outside view ofthe Earth-Moon space and would thus be well-suited for the placement of navigation or telecommunicationconstellations in support of continued human presence on the Moon. The analysis of such mission architecturewill be considered in future research.

The second class of cyclers considered were organized around the notion of connections associated withunstable periodic orbits of the third species. More precisely, we illustrated the use of homoclinic connectionsto the L1 Lyapunov family of periodic orbits. These lower energy cyclers provide more flexibility in modifyingthe orbit characteristics, such as the frequency of fly-bys or the Earth or Moon and may form the basis ofuseful mission architectures for the servicing to libration point missions or the periodic transport of cargoto the Moon.

While the present work presented several examples of cyclers and emphasized the methodology, furtherexploration of the stability of such cyclers, as well as the design of sample mission architectures based onthese trajectories require additional work. In particular, the optimization of the criteria mentioned in thispaper to design constellations of telecommunication or navigation spacecraft covering the whole Earth-Moonsystem would be of interest. Also, the current work focused on the planar case, and future research shouldaddress the cyclers in the three-dimensional problem and account for solar perturbations.

Acknowledgments

The research described in this paper was financially supported by the California-Catalunya EngineeringInnovation Program. Our industrial partners, M. D. Talbot of Universal Space Lines, LLC, Newport Beach,California and Marta Escudero of GTD Enginyeria de Sistemes i de Software, Barcelona, Spain, are gratefullyacknowledged.

References

1Broucke, R. A., Periodic Orbits in the Restricted Three-body Problem with Earth-Moon Masses, Jet Propulsion Labora-tory, California Institute of Technology, Pasadena, 1968.

2Cooke, D. R., Joosten, B. K., Lo, M. W., Ford, K. M., and Hansen, R. J., “Innovations in Mission Architectures forExploration Beyond Low Earth Orbit,” Acta Astronautica, Vol. 53, No. 4, August 2003, pp. 387–397.

3Belbruno, E., “A Low Energy Lunar Transportation System Utilizing Chaotic Dynamics,” AAS/AIAA AstrodynamicsSpecialists Conference, Lake Tahoe, California, August 7-11 2005, pp. 1–7.

4Johnson, M. D. and Belbruno, E. A., “Reduction of Lunar Landing Fuel Requirements by Utilizing Lunar BallisticCapture,” New Trends in Astrodynamics and Applications - Annals of the New York Academy of Sciences, Vol. 1065, 2005,pp. 139–151.

5Belbruno, E. A., “Low-Energy Pathways in Space,” American Scientist , Vol. 94, No. 5, September-October 2006, pp. 390.6Russell, R. P. and Strange, N. J., “Planetary Moon Cycler Trajectories,” AAS/AIAA Space Flight Mechanics Meeting,

Sedona, Arizona, January 28 - February 1 2007, pp. 1–20.7Russell, R. P., “Global Search for Planar and Three-Dimensional Periodic Orbits Near Europa,” The Journal of the

Astronautical Sciences, Vol. 54, No. 2, April-June 2007, pp. 199–226.8Barrabes, E. and Gomez, G., “Spatial p-q Resonant Orbits of the RTBP,” Celestial Mechanics and Dynamical Astronomy,

Vol. 84, No. 4, December 2002, pp. 387–407.9Barrabes, E. and Gomez, G., “Three-Dimensional p-q Resonant Orbits Close to Second Species Solutions,” Celestial

Mechanics and Dynamical Astronomy, Vol. 85, No. 2, February 2003, pp. 145–174.10McGehee, R. P., Some Homoclinic Orbits for the Restricted Three-Body Problem, Ph.D. thesis, University of Wisconsin,

1969.11Llibre, J., Martinez, R., and Simo, C., “Tranversality of the Invariant Manifolds Associated to the Lyapunov Family of

Periodic Orbits Near L2 in the Restricted Three-Body Problem,” Journal of Differential Equations, Vol. 58, No. 1, October1985, pp. 104–156.

12Howell, K. C., Barden, B. T., Wilson, R. S., and Lo, M. W., “Trajectory Design Using a Dynamical Systems Approachwith Application to GENESIS,” Advances in the Astronautical Sciences, Vol. 97, No. 1, October 1998, pp. 1665–1684.

19 of 20

American Institute of Aeronautics and Astronautics

13Koon, W. S., Lo, M. W., Marsden, J. E., and Ross, S. D., “Low Energy Transfer to the Moon,” Celestial Mechanics andDynamical Astronomy, Vol. 81, No. 1-2, September 2001, pp. 63–73.

14Dunham, D. W. and Farquhar, R. W., “Libration Point Missions, 1978-2002,” Libration Point Orbits and Applications,2003.

15Gomez, G., Koon, W. S., Lo, M. W., Marsden, J. E., Masdemont, J., and Ross, S. D., “Connecting Orbits and InvariantManifolds in the Spatial Restricted Three-Body Problem,” Nonlinearity, Vol. 17, No. 5, May 2004, pp. 1571–1606.

16Lo, M. W. and Parker, J. S., “Unstable Resonant Orbits Near Earth and Their Applications in Planetary Missions,”AIAA/AAS Astrodynamics Specialists Conference and Exhibit , Providence, Rhode Island, August 16-19 2004, pp. 1–29.

17Parker, J. S. and Lo, M. W., “Shoot the Moon 3D,” Advances in the Astronautical Sciences, Vol. 123, No. 1, October2006, pp. 2067–2086.

18Leiva, A. M. and Briozzo, C. B., “The Earth-Moon CR3BP: A Full Atlas of Low-Energy Fast Periodic Transfer Orbits,”2006.

19Lara, M., Russell, R. P., and Villac, B. F., “Fast Estimation of Stable Regions in Real Models,” Meccanica, Vol. 42,No. 5, October 2007, pp. 511–515.

20Lara, M., Russell, R. P., and Villac, B. F., “Classification of the Distant Stability Regions at Europa,” Journal ofGuidance, Control, and Dynamics, Vol. 30, No. 2, March-April 2007, pp. 409–418.

21Allgower, E. L. and Georg, K., Introduction to Numerical Continuation Methods, Springer-Verlag, Berlin, 1990.22Simo, C., “On the analytical and numerical approximation of invariant manifolds,” Modern methods in Celestial Me-

chanics, edited by D. Benest and C. Froeshle, Editions Frontieres, 1990, pp. 285–330.23Doedel, E. J., Paffenroth, R. C., Keller, H. B., Dichmann, D. J., Galan-Vioque, J., and Vanderbauwhede, A., “Compu-

tation of periodic solutions of conservative systems with application to the 3-body problem,” Internat. J. Bifur. Chaos Appl.Sci. Engrg., Vol. 13, No. 6, 2003, pp. 1353–1381.

24M.Henon, Generating families in the restricted three-body problem, Springer, 1997.25Canalias, E. and Masdemont, J. J., “Homoclinic and heteroclinic transfer trajectories between planar Lyapunov orbits

in the sun-earth and earth-moon systems,” Discrete Contin. Dyn. Syst., Vol. 14, No. 2, 2006, pp. 261–279.26Barrabes, E., Mondelo, J. M., and Olle, M., “Numerical Continuation of Families of Homoclinic Connections of Periodic

Orbits in the RTBP,” In preparation, 2008.27B.F.Villac and D.J.Scheeres, “On the concept of periapsis in Hill’s problem,” Celestial Mechanics and Dynamical As-

tronomy, Vol. 90, 2004.

20 of 20

American Institute of Aeronautics and Astronautics