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AAS 21-625

USING FINITE-TIME LYAPUNOV EXPONENT MAPS FOR

PLANETARY MOON-TOUR DESIGN

David Canales∗

, Kathleen C. Howell†

, and Elena Fantino‡

The focus of the present investigation is an efﬁcient and general design strategy for

transfers between planetary moons that fulﬁll speciﬁc requirements. The strategy

leverages Finite-Time Lyapunov Exponent (FTLE) maps within the context of the

Moon-to-Moon Analytical Transfer (MMAT) scheme previously proposed by the

authors. Incorporating FTLE maps with the MMAT method allows direct transfers

between moons that offer a wide variety of trajectory patterns and endgames de-

signed in the circular restricted three-body problem, such as temporary captures,

transits, takeoffs and landings. The technique is applicable to several mission sce-

narios, most notably the design of a moon tour.

INTRODUCTION

Many space missions focus on the exploration of planetary moons, such as those of the gas

giants or Mars. These missions sometimes include orbiters or surface landers for key moons. For

example, JAXA’s MMX1launching in 2024 foresees the return of a Phobos sample to Earth.2Some

near-term NASA missions are also planned, e.g., Europa Clipper3for the exploration of Europa

and Dragonﬂy4with a goal to land a robot on Titan. The satisfaction of the increasingly complex

mission objectives imply an understanding of multi-body dynamics in the vicinity of the target

moons. Furthermore, given the challenges in reaching these systems from Earth, multi-moon tours

are an appealing option allowing the exploration of different objects within the same mission.

Numerous studies have been devoted to the construction of transfers between planetary moons

in multi-body environments. It has been demonstrated by various authors that low-energy transfers

between moons orbiting a common planet can be successfully designed using invariant manifolds.

Some authors tackle the problem using the coupled circular restricted three-body problem (CR3BP)

and identify connections between the moons using Poincaré sections.5, 6 Strategies involving moon

ﬂybys are examined using the Tisserand graph in the two-body problem (2BP),7,8 as well as the

CR3BP9and the patched 2BP-CR3BP.10, 11 The moon-tour design process using these approaches

becomes challenging when more complex elements such as libration point orbits, captures, land-

ings or even the return to a previously visited moon are included in the mission scenario. Other

techniques12–15 are based on trajectories departing and approaching distinct moons in the form of

∗Ph.D. Candidate, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907;

dcanales@purdue.edu

†Hsu Lo Distinguished Professor of Aeronautics and Astronautics, School of Aeronautics and Astronautics, Purdue Uni-

versity, West Lafayette, IN 47907; howell@purdue.edu

‡Assistant Professor, Aerospace Engineering Department, Khalifa University of Science and Technology, P.O. Box

127788, Abu Dhabi, United Arab Emirates; elena.fantino@ku.ac.ae

1

conic arcs; any available connections are then determined analytically, producing an optimal rel-

ative phase between the moons as well as an analytical minimum-∆vtransfer. While most of

these investigations assume that the moons are in coplanar orbits, Canales et al., 2020, introduce

the Moon-to-Moon Analytical Transfer (MMAT) method,15 proposed as a useful strategy to design

transfers between planar as well as spatial periodic orbits associated with two moons located in their

true orbital planes. The MMAT method also proves useful for computing transfers between differ-

ent types of resonant orbits in the Martian system16 where, due to the small masses of Phobos and

Deimos, the invariant manifolds emanating from libration point orbits do not intersect in conﬁgu-

ration space. By means of Mars-Deimos resonant orbits in the Mars-Deimos CR3BP, the MMAT

approach locates potential transfers to the vicinity of Phobos.16

Various authors have described the behavior of trajectories departing or approaching the vicinity

of a moon in a multi-body environment. The type of motion exhibited in the vicinity of a moon

is closely related to the energy level of the trajectory. The scalar ﬁeld associated with the Finite-

Time Lyapunov Exponent (FTLE) is employed to locally measure the largest stretching direction

along the ﬂow of a trajectory17–19 consistent with the search for hyperbolic Lagrangian Coherent

Structures,20, 21 that deﬁne the extension of regions bounding different behaviors. The objective

in incorporating FTLE maps is the identiﬁcation of useful paths emerging from an initial point

and, by leveraging the largest stretching direction along the ﬂow, the parameterization of regions

with distinct types of ﬂow. It has been demonstrated that FTLE maps originating from Poincaré

sections near the L1or L2gateways in the CR3BP constitute a useful resource to identify families

of trajectories transiting through the vicinity of a moon. Building on this concept, FTLE maps have

been employed to design trajectories that approach Oberon and Titania in the Uranus system and

are characterized by a desired behavior.22 These low-energy trajectories have been produced in the

coupled CR3BP assuming that the moons revolve on coplanar orbits.

Finite-Time Lyapunov Exponent maps (FTLE maps) offer insight into captures, departures and

impacts with a moon with good computational performance. The present investigation merges the

beneﬁts offered by FTLE maps with the MMAT methodology. The objective is the design of low-

energy transfers between moons with speciﬁc behaviors in the vicinity of the departure and desti-

nation bodies. The technique accommodates the true orbital planes of the moons and is applicable

to any planetary system. In the vicinity of the moons, the use of dynamical structures in the three-

body problem is examined. As a result, by establishing a relationship between FTLE maps and the

MMAT method, a design process emerges to determine a trajectory with a given behavior departing

one moon that yields a direct transfer towards a capture or a transit orbit at the destination moon.

Then, given the analytical basis of the MMAT approach, it is possible to quickly determine the

cost, transfer time, and required relative phases between the moons, thus, enabling a transfer. In

addition, the strategy can be formulated in two ways: (a) to determine all the arrival trajectories at a

given energy level accessible from a selected departure trajectory; or (b) to identify all the departure

trajectories at a given level of energy that deliver access to a desired arrival trajectory.

The dynamical model in this investigation is ﬁrst introduced. Then, a brief theoretical background

is provided about the MMAT method and the mathematics behind FTLEs. Additionally, a brief

explanation on the computation of FTLE maps and their particularities is included. The Moon-

to-Moon Access Maps methodology is then presented, one that leverages both the MMAT scheme

with the FTLE maps to plan moon-to-moon transfers that offer particular behaviors in the moons’

vicinities.

2

DYNAMICAL MODEL

The design of trajectories in multi-moon systems accommodates the gravitational accelerations of

several bodies simultaneously acting on the spacecraft (s/c). The classical CR3BP incorporates only

two bodies affecting the dynamics of the s/c and, thus, it is sometimes challenging to construct valid

preliminary trajectories to travel from one moon to another. For the purposes of this investigation,

a spatial 2BP-CR3BP patched model is employed to design transfers between the moons. The s/c

trajectory within the vicinity of the moons is modeled using the planet-moon CR3BP problem.

Outside the moons’ vicinities, the s/c motion is designed using the 2BP with the focus at the planet.

When analyzing the s/c motion in the vicinity of a moon, the CR3BP23 is a simpliﬁed model that

offers useful insights for many different applications. In this investigation, the CR3BP describes the

motion of a s/c subject to the gravitational forces of the planet and the moon, that are both assumed

to move in circular orbits about the center of mass of the system. To model the problem, a system of

differential equations is written in dimensionless form. The mass ratio of the system, µ, is deﬁned as

µ=mm/(mm+mp), with mmand mpbeing the masses of the moon and the planet, respectively.

A barycentric rotating frame is conveniently used to represent the motion of the s/c, with the ˆx-

axis directed from the planet-moon barycenter to the moon and the ˆz-axis from the barycenter in

the direction of the orbital angular momentum vector. In this frame, the planet and the moon are

located at positions ¯rp= [−µ, 0,0]Tand ¯rm= [1 −µ, 0,0]T, respectively. Note that overbars

denote vectors whereas the superscript ’T’ indicates the transpose of a vector. The evolution of the

s/c position ¯rrot = [x, y, z]Tand velocity ˙

¯rrot = [ ˙x, ˙y, ˙z]Tis governed by the following equations

of motion:

¨x−2 ˙y=∂U∗

∂x ; ¨y+ 2 ˙x=∂U ∗

∂y ; ¨z=∂U∗

∂z (1)

where dots indicate derivatives with respect to dimensionless time, and U∗=1−µ

rp−s/c +µ

rm−s/c +

1

2(x2+y2)represents the pseudo-potential function for the system of differential equations (rp−s/c

and rm−s/c are the distances of the s/c to the planet and the moon, respectively). Additionally,

ﬁve equilibrium solutions exist in the given formulation when both the velocity and acceleration of

the s/c are null. Such equilibrium solutions are denoted as the libration points (L1to L5). Given

the linear stability properties of the libration points, motion exists categorized by different types of

families of periodic and quasi-periodic orbits.24, 25 The family of periodic orbits in this investigation

is the planar Lyapunov orbit family. Hyperbolic invariant manifolds that emanate from periodic

orbits serve as pathways in the vicinity of the moons and between periodic orbits within the same

system.26 Unstable manifolds depart the vicinity of a periodic orbit and stable manifolds arrive in

the vicinity of a periodic orbit. In this investigation, transit orbits are trajectories that enter or leave

the vicinity of the moon through an L1or L2gateway and revolve around the moon before colliding

or again departing the moon vicinity. Note that the planet and the moons are assumed perfectly

spherical and, thus, other harmonics of the motion caused by irregularities in the gravity ﬁeld are

neglected. Finally, note that the Jacobi constant (JC ) is deﬁned as the "energy" of the s/c in the

given CR3BP system via

JC = 2U∗−( ˙x2+ ˙y2+ ˙z2).(2)

Trajectories within the vicinity of a moon that are propagated in the CR3BP are bounded in conﬁgu-

ration space by the Sphere of Inﬂuence (SoI) of the moon. When either departure or arrival CR3BP

trajectories cross the SoI, the motion is modeled in terms of the planet-2BP with a focus at the planet

and uniquely determined by the osculating orbital elements: semi-major axis (a), eccentricity (e),

3

right ascension of the ascending node (Ω), inclination (i), and argument of periapsis (ω), retrieving

also the true anomaly (θ) of the s/c. A schematic for the patched 2BP-CR3BP is represented in Fig-

ure 1 (trajectories emanating from the departure and arrival moons are in different planes). Given

an assumption that, beyond both moons’ SoIs the motion is considered Keplerian, it is feasible to

complete an analytical exploration for a connection in space between departure and arrival conics

in the inertial frame. To deﬁne the radius of the SoI for the moons, the deﬁnition appears in Canales

et al., 2021.27 It is speciﬁed as the distance from the moon along the ˆx-rotating axis to the point

for which the ratio aSoI =am

apis equal to a certain small amount (amand apare the gravitational

accelerations of the moon and the planet, respectively). The ratio aSoI is, thus, a free parameter for

the design of the transfers. Since the transfers are determined using the Keplerian conics beyond

the SoI, the selection of aSoI modiﬁes the resulting moon-to-moon transfer, under-predicting ∆v’s

or disregarding shorter time-of-ﬂight solutions.27

Figure 1. Schematic of the blending of two different systems in the planet-centered

inertial frame using the spatial 2BP-CR3BP patched model.

THEORETICAL BACKGROUND

To design low-energy transfers between moons that satisfy speciﬁc behaviors, this investigation

merges the beneﬁts offered by the FTLE maps and the MMAT scheme. First, a brief insight on both

methodologies is provided with their beneﬁts for the overarching goal of this investigation. For

explanatory purposes, the analysis is based on an example to explore transfers from Ganymede to

Europa. Here, aS oI is considered the same for both moons and equal to aSoI = 5·10−4.27 However,

the methodology is applicable for transfers between any two consecutive moons where the relative

distance between moon orbits allows.

The MMAT method

The objective of the analysis is the design methodology for transfers between the neighborhood

of individual consecutive moons in which the departure and arrival trajectories are solutions in the

respective planet-moon CR3BPs. The actual inclinations of the moon orbital planes are incorpo-

rated and the transfer develops in three dimensions (3D). A large number of trajectories depart and

4

Figure 2. Constructing a direct transfer scenario between two consecutive moons

using the MMAT method.

approach the vicinity of the two moons, but the optimal solution depends on their relative orbital

phase and is epoch-dependent due to the 3D geometry of the problem.

A methodology previously introduced by the authors,27 i.e., the MMAT method, initially lever-

ages some simpliﬁcations to yield lower costs and shorter times-of-ﬂight for transfers assuming that

both moon orbits are in their true orbital planes. The method relies on the 2BP-CR3BP patched

model. Consequently, the planet-moon CR3BP trajectories are initially approximated with conic

arcs far from the moons. For example, in the design of a transfer from Ganymede to Europa, the

departure Jupiter-Ganymede (J-G) CR3BP orbits are approximated by departure conics, whereas

the arrival Jupiter-Europa (J-E) CR3BP orbits are represented as arrival conics. The two types of

Keplerian orbits include a common focus at Jupiter. Given the two Keplerian orbits, it is possible to

analytically explore promising trajectories and conﬁgurations between the moons (Figure 2 offers a

schematic for the concept). The core of the MMAT method is a necessary analytical condition for a

given arrival conic to deliver a spatial intersection (corresponding to an impulsive velocity variation

∆V) with a given departure conic.15 Such a condition reads:

aa(1 −ea)≤ad(1 −e2

d)

1 + edcos(θdInt +nπ)≤aa(1 + ea),with n= 0,1,(3)

where aaand adare deﬁned as the semi-major axes of the arrival and departure conic, respectively,

eaand edare their eccentricities, whereas the true anomalies θdInt and θdI nt +πof the two points of

intersection depend on the inclination and right ascension of the ascending nodes of the departure

and arrival conics, as well as on the argument of the periapsis of the departure conic (that, in turn, de-

pends on the departure epoch θ0Gan at the origin of the transfer with respect to the departure moon).

If the above condition is satisﬁed, it is possible to phase the conics (hence, determine a suitable

relative orbital phase between the moons) to enable such a transfer, and determine its performance

in terms of total transfer time ttot and ∆v. In summary, the MMAT technique identiﬁes and fully

characterizes one-impulse moon-to-moon low-energy transfers given the epoch of departure from

one of the moons.

Finite-Time Lyapunov Exponents

Dynamical systems theory leverages different techniques with two-dimensional Poincaré maps

to investigate the long-term dynamics in the CR3BP. The Cauchy-Green Strain Tensor (CGST) is

5

widely used to understand ﬂow characteristics given the evolution of a perturbation vector over time.

To introduce the CGST, the state transition matrix (STM) is ﬁrst introduced. The STM, φ(tf, t0),

relates the ﬂow in the vicinity of a trajectory via a linear mapping correlating the initial perturbation,

δ¯x0, to the ﬁnal perturbed states downstream, δ¯xf:

δ¯xf=φ(tf, t0)δ¯x0,(4)

where t0and tfare the initial and ﬁnal times along the propagated trajectory, respectively. Pertur-

bations can exhibit distinct patterns of divergence, compression or lack of either. Using a singular

value decomposition of the STM, it is possible to characterize the behavior of these perturbations.

The singular values, σi, encode such behavior, since the contraction or expansion of the local phase

is given by σiin a direction deﬁned by Vi(direction of stretching at t0). Using a singular value

decomposition, the STM is decomposed into:

φ(tf, t0) = UΣVT,(5)

where the columns of Urepresent the direction of stretching at the ﬁnal epoch and Σis a diagonal

matrix with the magnitude of stretching along different directions in ascending order (σ1> σ2>

... > σn):

Σ =

σ10... 0

0σ2... 0

.

.

..

.

.....

.

.

0 0 ... σn

.(6)

Note that the matrices Vand Uare mutually orthogonal. Figure 3 reveals a simpliﬁed 2-D schematic

of the SVD that reﬂects the contraction or expansion of the ﬂow, as well as the most and least

stretching directions, revealing the directions more and less sensitive to perturbations, respectively.

Similarly, the CGST deﬁnes the deformation as the product of the transpose of the state transition

matrix with itself:28

C(tf, t0) = φT(tf, t0)φ(tf, t0).(7)

From the eigendecomposition of C, it is possible to produce the eigenvalues of C,λ, related to Σ

as λi=σ2

i. The eigenvectors of Care the same as U. Then, given that the largest singular value σ1

corresponds to the largest possible growth of a perturbation, the Finite-Time Lyapunov Exponent

(FTLE) is deﬁned as:

F T LE =σ1

|∆t|(8)

where ∆tcorrespond to the propagation time: ∆t=tf−t0. Of course, there exists an FTLE for

every σi, but the FTLE computed in Eq. (8) corresponds to the largest one since it is the one that

represents the maximum stretching. As a result, FTLEs measure the relative growth or contraction

of the phase-space elements under the inﬂuence of the system ﬂow over a given time interval.

FTLE maps

Once FTLEs are deﬁned, their beneﬁts are apparent when represented in terms of Poincaré sections.

Finite-Time Lyapunov Exponent maps (FTLE maps) supply quantitative information about the type

of motion that develops during trajectory propagation at a given energy level over a speciﬁc time

interval following departure from or prior to the approach in the vicinity of a moon. Sample be-

haviors include: (a) captured trajectories (captures); (b) collision trajectories (landings or takeoffs);

6

Figure 3. Stretching associated with eigenvector of the Cauchy-Green Strain Tensor.

(c) trajectories that depart the vicinity of the moon (tours). It also separates the ﬂow entering to-

wards the vicinity of the moon through the corresponding gateway versus one that does not. For

the sake of illustration, consider the previous transfer from Ganymede to Europa. Let the departure

from Ganymede and arrival to Europa correspond to trajectories with Jacobi constant values J Cd=

3.00754 and JCa=3.00240, respectively. Two FTLE maps are produced, one for departure (Fig-

ure 4) and one for arrival (Figure 5). The former accommodates the trajectories departing the vicin-

ity of Ganymede through a Poincaré section positioned just beyond the J-G L1gateway (Σdeparture

in Figure 4(a)); it is constructed by backward propagating all the state vectors on the section over

a certain nondimensional time interval td(td=10 equals 11.4 days in normalized units of the J-G

CR3BP). Similarly, the arrival FTLE map (Figure 5) describes the trajectories approaching Europa

through a Poincaré section positioned just outside the J-E L2gateway (Σarrival in Figure 5(a)) and

corresponds to forward propagations of all the states on the section over a time ta(ta= 10, that is

5.65 days in normalized units of the J-E CR3BP). Note that, in both maps (Figure 4(b) and 5(b)),

the trajectories that transit the vicinity of the moons are maintained within the relevant (i.e., unstable

for J-G CR3BP, stable for the J-E CR3BP) hyperbolic invariant manifolds associated with the planar

Lyapunov orbits at the given Jacobi constant levels around L1and L2, respectively. The different

streamlines appearing in the maps mark initial conditions corresponding to different behaviors, i.e.,

different outcomes or motion patterns or endgames, as illustrated in Figure 6. Therefore, FTLE

maps are useful as employed in a selection process for the initial conditions that deliver a transfer

with speciﬁed characteristics at departure and arrival. This fact is achieved by the combination of

departure and arrival FTLE maps and their integration with the MMAT method.

THE MOON-TO-MOON ACCESS MAPS

Incorporating FTLE maps with the MMAT method allows direct transfers between moons that

offer a wide variety of trajectory patterns and endgames. Consider the transfer from Ganymede to

Europa. A Jacobi constant value for departure from Ganymede and one for arrival to Europa is ﬁrst

selected to compute the FTLE maps in Figures 4 and 5, respectively. Transfers between two distinct

moons are epoch dependent,27 specially considering that moons are in their true orbital planes.

To compute the aforementioned maps and evaluate Eq. (3), it is important to select the departure

epoch from the departure moon that allows a large number of available transfers, assuming any

exists. For an example, consider that the departure epoch from Ganymede is θ0Gan = 82.506◦

(previous simulations suggest that this value leads to the largest number of available connections

between Ganymede and Europa for the speciﬁed departure and arrival energy levels). First, the

initial conditions on the departure map (Σdeparture) are propagated towards the sphere of inﬂuence

7

(a) Schematic of trajectories departing the Ganymede

vicinity through L1.

(b) Departure FTLE map that includes all

trajectories at a J Cd= 3.007538 departing

the Ganymede vicinity with a non-dimensional

propagation time of td=−10. Note that the

non-dimensional units here correspond to the

Jupiter-Ganymede CR3BP.

Figure 4. Deﬁnition of the departure FTLE map.

(a) Schematic of trajectories arriving in the Europa

vicinity through L2.

(b) Arrival FTLE map that includes all trajectories at

aJ C=3.0024 arriving towards the Europa vicinity with

a non-dimensional propagation time of ta= 10. Note

that the non-dimensional units here correspond to the

Jupiter-Europa CR3BP.

Figure 5. Deﬁnition of the arrival FTLE map.

8

Figure 6. Some departure trajectory behaviors as a function of the initial condition

obtained from the departure FTLE map.

(SoI) of the departure moon (Ganymede in the example). Since arcs are evaluated as conics in

the inertial Jupiter-centered 2BP beyond the SoI, the central expression in Eq. (3) is evaluated at

the section for every departure conic. The resulting map is named as the Moon-to-Moon Tides

map (Figure 7 (a)), where the colors reﬂect the value of 1

L∗

a

ad(1−e2

d)

1+edcos(θdInt )(denoted, hereafter, as

the "θdInt conﬁguration") for each initial condition propagated from the Poincaré section towards

the Ganymede’s SoI. Note that the shape of the Moon-to-Moon Tides map (Figure 7 (a)) and the

departure FTLE maps (Figure 4) is the same one, but the ﬁrst one propagates initial conditions

towards the SoI and computes 1

L∗

a

ad(1−e2

d)

1+edcos(θdInt ), whereas the latter propagates initial conditions

towards the Ganymede vicinity by computing the FTLEs. Simultaneously, initial conditions on

the arrival map (Σarrival) are propagated backwards in time towards the destination moon’s SoI

(here Europa), where the right and left expressions in Eq. (3) (aa(1 + ea)/L∗

aand aa(1 −ea)/L∗

a,

respectively) are evaluated at the section for each arrival conic. The resulting maps are denoted as

the Upper- and Lower-Constraint Arrival maps (Figure 7 (b) and Figure 7 (c), respectively). Note

that the characteristic length L∗

ain the arrival planet-moon CR3BP is employed to represent the three

maps in Figure 7 to the same scale. Finally, these are overlapped (interpolated) with the information

in the Moon-to-Moon Tides map in Figure 7 to assess the available transfer trajectories between the

moons. This procedure yields results for available transfers between departure and arrival moons

as represented in the schematic in Figure 8, in which the grey region corresponds to Ganymede-to-

Europa transfers that satisfy Eq. (3). The resulting methodology that relates the vicinities of two

moons to locate available transfers that satisfy speciﬁc behaviors is denoted hereafter the Moon-to-

Moon Access Maps method (MMAT Maps in brief). The 1.242 isoline is employed in Figure 7 as a

9

Figure 7. Interpolation between Moon-to-Moon Tides map (left) and Upper- and

Lower-Constraint Arrival maps (upper-right and lower-right, respectively) for com-

puting transfers for the Ganymede to Europa journey. The 1.242 isoline is used as a

reference value to evaluate possible connections between the maps.

Figure 8. Sketch of the MMAT Maps method.

reference to explore the overlap of the three maps, i.e.,

ad(1 −e2

d)

1 + edcos(θdInt )=aa(1 + ea)=1.242 ×L∗

a.(9)

Note that the central expression in Eq. (3) is always smaller or equal to aa(1 + ea)for a transfer

to occur. In this example, the Moon-to-Moon Tides map with the Upper-Constraint Arrival map is

analyzed because the transfer is from an outer moon to an inner moon. For this case, the inequality

aa(1 −ea)≤ad(1−e2

d)

1+edcos(θdInt +nπ)is always satisﬁed, which is the main reason why Figure 7 (c)

is all yellow (the values are much smaller than the ones in the Moon-to-Moon Tides map, thus,

10

always thus fulﬁlling the lower inequality). Therefore, it is important to ensure that trajectories

approaching in the arrival moon vicinity possess an apoapsis larger than the middle term of Eq.

(3). However, for a journey from an inner towards an outer moon, the Moon-to-Moon Tides map

is compared with the Lower-Constraint Arrival map since it is mandatory that ad(1−e2

d)

1+edcos(θdInt +nπ)

is always larger than the periapsis of the arrival conic. Thus, for inner-to-outer moon transfers, the

Upper-Constraint Arrival map (7 (b)) possesses a uniform color. Given the nature of the dynamics of

a trajectory departing an inner moon vicinity and without an extra impulse, it is not feasible that the

apoapsis of the departure conic is greater than the apoapsis of a conic arriving towards the vicinity

of an outer moon. Returning to the sample scenario from Ganymede to Europa, the red arrows

in Figure 7 indicate the set of initial conditions (i.e., fulﬁlling Eq. (3)) available for selection that

yield one-impulse trajectories from Ganymede to Europa. Under these conditions, an arrival epoch

exists such that departure and arrival trajectories connect in space and, thus, a successful transfer

between the moons is produced. In addition, the isolines are projected onto the departure and arrival

FTLE maps (Figure 9), and allow inspection for distinct available transfers between Ganymede and

Europa leveraging the streamlines that offer different behaviors inside the FTLE maps. Therefore,

it is possible to select a feasible transfer between trajectories exhibiting desired behaviors in the

vicinity of each moon. For example, the selected initial conditions for the departure and arrival

FTLE maps in Figure 9 lead to the sample transfer plotted in Figure 10 in which the spacecraft,

after completing some revolutions around Ganymede and with a single impulsive ∆v, transits to the

vicinity of Europa towards the interior region of the J-E CR3BP.

Figure 9. Available transfers connecting Ganymede and Europa between departure

(left) and arrival (right) FTLE maps. The 1.242 isoline is used as a reference value to

see where both maps equate.

Cost and arrival moon phase inspection maps for moon-to-moon transfers

The MMAT Maps incorporating FTLE information are a useful resource to design different types

of transfers between distinct moons. In summary, selecting a Jacobi constant for departure and

arrival, it is possible to straightforwardly inspect the trajectories and determine those that deliver a

tour between the two moons; the resulting conﬁguration is then quickly retrieved. For more speciﬁc

applications, it is possible to select trajectories (as in Figure 9) to construct a tour that satisﬁes certain

11

Figure 10. Transfer trajectory from Ganymede to Europa given the desired behav-

iors from the FTLE maps. The transfer is represented in the Jupiter-centered Ecliptic

J2000 frame (left), Jupiter-Ganymede rotating frame (top-right) and Jupiter-Europa

rotating frame (bottom-right).

objectives. However, the MMAT scheme is further leveraged to obtain a visual representation for the

total ∆vbudget, the transfer time, and the required relative phases between moons that enable the

transfers. Two design options are possible: (a) selection of all the departure trajectories at a given

level of energy that deliver access to a desired arrival trajectory; or (b) identiﬁcation of all the arrival

trajectories at a given energy level accessible from a selected departure trajectory. In this section,

option (a) is further analyzed. Consider that the target trajectory in the Europa vicinity is the stable

manifold trajectory that arrives asymptotically towards the L2Lyapunov orbit (JCa=3.00240).

The selected trajectory is then the ﬁrst stable manifold trajectory that is available and accessible

from Ganymede. In the Upper-Constraint Arrival map, it corresponds to the manifold trajectory

that intersects with the isoline equal to aa(1+ea)

L∗

a= 1.2750 (Figure 11). Assume that the departure

epoch from Ganymede is again θ0Gan = 82.506◦. After applying the MMAT method to the initial

conditions in the departure FTLE map (Figure 4), the total ∆vbudget, the transfer time, and the

required relative phases between moons is available for all the conditions where a transfer is possible

(Figure 12). The cost and phase inspection maps in Figure 12 provide quantitative information such

that initial conditions in the departure map satisfy certain conditions within reasonable cost budgets.

For example, selecting the trajectory in Figure 12(a), a trajectory that completes close passages of

the moon before departing is selected and the resulting total transfer is represented in Figure 13.

Similarly, the same analysis is performed for design option (b). If the s/c currently moving along

a speciﬁc trajectory and completing several revolutions around Ganymede, it is possible to apply

MMAT and determine all available trajectories upon arrival to the Europa vicinity and to explore

their associated costs. Consequently, similar maps to those in Figure 12 are produced but now

leveraging the arrival FTLE map in Figure 5. For a sample result, Figure 14 illustrates a transfer

constructed using this methodology from Ganymede to plan a landing on Europa: after several

revolutions around Ganymede, the s/c eventually reaches the surface of Europa.

12

(a) Upper-Constraint Arrival map (isoline at 1.2750).

(b) Selected arrival trajectory that targets an L2Lya-

punov orbit in the J-E CR3BP with a JCa=3.00240.

Figure 11. Selected arrival condition towards the Europa vicinity.

(a) Available departure trajectories

from Ganymede. (b) Total ∆vbudget.

(c) Total ttot budget. (d) Phase of Europa at its plane.

Figure 12. Available departure trajectories and associated costs from Ganymede with

JCd= 3.00754 and access along the arrival trajectory to Europa at θ0Gan = 82.506◦

(the black line outlines the unstable manifold bounding arrival options in the map).

13

Figure 13. Transfer from Ganymede to Europa that arrives along an L2Lyapunov

orbit in the J-E system after providing close passages of Ganymede. The transfer is

represented in the Jupiter-centered Ecliptic J2000 frame (left), J-G rotating frame

(top-right) and J-E rotating frame (bottom-right).

Figure 14. Transfer from Ganymede to Europa that is initiated with several rev-

olutions around Ganymede before arrival at the surface of Europa. The transfer is

represented in the Jupiter-centered Ecliptic J2000 frame (left), J-G rotating frame

(top-right) and J-E rotating frame (bottom-right).

14

Moon-to-moon access dependence

The analysis completed using the MMAT Maps yields an apparent relationship between the num-

ber of trajectories that arrive in a moon vicinity and the volume of trajectories that depart another

moon. Consider again the transfer from Ganymede (Figure 4) to Europa (Figure 5) with a depar-

ture epoch θ0Gan = 82.506◦. Recall that the isolines in the MMAT Maps correspond to the value

where both ad(1−e2

d)

1+edcos(θdInt )and aa(1 + ea)equate to a certain value (or ad(1−e2

d)

1+edcos(θdInt )=aa(1 −ea)

for transfers from inner to outer moons). Observe in Figure 15 that shifting the 1.242 isoline value

to 1.26 offers a larger number of initial conditions in the Moon-to-Moon Tides map at the expense

of fewer available initial conditions in the Upper-Constraint Arrival map, and vice versa. This fact

translates into the following conclusions: (a) Increased availability of options for transit orbits and

unstable manifold trajectories to depart one moon is offset by the number of feasible trajectories

that are available for arrival at another moon by means of a direct moon-to-moon transfer; and (b)

for a greater number of options to reach the arrival moon, the number of conditions (i.e., trajectory

behaviors) for which the s/c is allowed to depart the departure moon are reduced for direct moon-to-

moon transfers. Consequently, an important inverse relationship occurs for planning moon-to-moon

transfers with a single ∆v(incorporating the fact that moons are located on their true orbital planes):

a larger volume of possible trajectories at departure of one moon imply fewer trajectory options for

arrival at another moon, and vice versa.

(a) Interpolation between Moon-to-Moon Tides map (left) and Upper-

Constraint Arrival map (right) incorporating a 1.242 isoline.

(b) Interpolation between Moon-to-Moon Tides map (left) and Upper-

Constraint Arrival map (right) incorporating a 1.26 isoline.

Figure 15. MMAT Maps applied to understand moon-to-moon access dependence

using two different isolines.

15

Moon-to-moon transfers dependence on epoch

Moon-to-moon transfers are epoch-dependent, implying that more or less feasible transfers as

observed on the MMAT Maps are available depending on the departure epoch from the departure

moon, i.e., Ganymede (θ0Gan ). Recall the creation process for the Upper- and Lower-Constraint

Arrival maps. They essentially represent the apoapsis and periapsis, respectively, along the arrival

conic as it crosses the arrival SoI (Figure 7). As a result, given that transit orbits entering the

vicinity of the arrival moon all ﬂow inside the stable manifold tubes, a speciﬁc stable manifold

trajectory is selected to aid in locating the best epochs for departure depending on the journey type:

(i) for transfers from an outer to an inner moon (i.e., Ganymede to Europa), the stable manifold

trajectory that leads to the maximum aa(1+ea)

L∗

ais selected; and (ii) for transfers from an inner to an

outer moon, the stable manifold trajectory that leads to the minimum value of aa(1−ea)

L∗

ais selected.

This ﬁrst maximum/minimum trajectory that allows access to the arrival moon is called hereafter

"minimum access arrival trajectory". The objective is a set of ’adequate’ epochs for departure

where a wider range of feasible transfer possibilities to access the minimum access arrival trajectory

are produced, i.e., more initial conditions within the departure FTLE map are available using the

isoline represented by (i) or (ii). Experience in the problem demonstrates that the epochs with

more available possibilities correspond to transfers between transit orbits and manifold trajectories

for two moons that include low ∆vpossibilities. Consider again the previous example; to inspect

the available connections from Ganymede to Europa, the minimum access arrival trajectory in this

example is the one that leads to the maximum aa(1+ea)

L∗

a(minimum access to Europa for the selected

JCa). Figure 16 illustrates the initial conditions in the departure FTLE map that vary depending on

θ0Gan for access to the Europa vicinity. For example, access to the Europa vicinity is not possible

when θ0Gan = 29◦. However, beyond θ0Gan = 35◦, more trajectories with distinct behaviors

become accessible until a maximum value at approximately θ0Gan = 82.506◦.

Accurately computing the Moon-to-Moon Tides map for every departure epoch to locate the

epoch with the maximum number of feasible connections between Ganymede and Europa is com-

putationally expensive. Consequently, some simpliﬁcations are employed. Consider the unstable

manifold departing the L1Lyapunov orbit from Ganymede in the previous example. For all the un-

stable manifold trajectories, the evolution of the middle term in Eq. (3) is computed for all unstable

manifold trajectories over θ0Gan = 0◦to 360◦, and it is evaluated with respect to the apoapsis and

periapsis along the minimum access arrival trajectory (see an example in Figure 17 for the epoch at

θ0Gan = 82.506◦). The angle on the horizontal axis in Figure 17 corresponds to the location of the

departure/arrival arc on the manifold along the periodic orbit, measured from the ˆx-axis.27 Once

the same analysis is completed for all θ0Gan values, the unstable manifold trajectory is selected that

possesses the minimum θdInt -conﬁguration for most of the epochs where the condition in Eq. (3) is

fulﬁlled. Then, the MMAT method is applied for the selected unstable manifold trajectory, resulting

in the "minimum access departure trajectory", for all θ0Gan to produce the ∆vand ttot budget for

this trajectory as accomplished in Canales et al., 2021.27 Then, the epoch with the lower ∆vvalue is

selected and the MMAT Maps are computed for the selected epoch, reducing the computation time.

Finally, for the minimum access departure trajectory in the current example, Figure 18 illustrates

the range of initial conditions upon the arrival FTLE map (and, thus, access towards the Europa

vicinity) and the variation depending on θ0Gan.

MMAT Maps for transfers between inner to outer moons

The main beneﬁt of the MMAT Maps approach is its versatility for applications in different sys-

tems, as well as journeys from outer to inner moons and vice versa. Consider now the reverse

16

Figure 18. Upper-Constraint Arrival map showing initial conditions (with access

from vicinity of Ganymede) upon arrival FTLE map depending on θ0Gan.

journey from that in the previous example: a tour to be designed from Europa (J Cd= 3.00240) to

Ganymede (JCa=3.00754). Since it is a transfer from an inner to an outer moon, Figure 19 rep-

resents the Moon-to-Moon Tides map and must be interpolated with the Lower-Constraint Arrival

maps. The 0.779 isoline is employed as a reference to explore the overlap of the two maps, i.e.,

ad(1 −e2

d)

1 + edcos(θdInt )=aa(1 −ea)=0.779 ×L∗

a.(10)

Therefore, the red arrows indicate the initial conditions from both departure and arrival FTLE maps

to offer selections for the design of tours from Europa to Ganymede. For a path from an outer to an

inner moon, the directions relative to the isolines are identiﬁed in Figure 7 and head towards the right

for both the departure and arrival FTLE maps. However, for transfers from inner to outer moons,

the initial conditions are selected in the opposite direction off the isoline for both the departure and

arrival FTLE maps (Figure 19).

SUMMARY

In this investigation, FTLE maps and the Moon-to-Moon Analytical Transfer method are com-

bined to produce accurate and realistic initial conditions for moon-to-moon transfer trajectories that

involve a single ∆vand satisfy speciﬁc behaviors in the vicinity of departure and arrival moons.

The proposed technique, i.e., the MMAT Maps, allows construction of moon-to-moon transfers that

satisfy a wide variety of different objectives in the vicinity of the moons, including trajectories

such that the spacecraft: (a) is inserted into periodic orbits, e.g., libration point orbits; (b) is grav-

itationally captured around the moon; (c) impacts with the lunar surface, enabling the addition of

takeoffs and landings in the moon-tour design; and (d) transits through the moon vicinity with a

potential connection with another moon vicinity. The method accommodates moons as located in

their true orbital planes. Note that moon-to-moon transfers designed using the MMAT scheme in

the spatial 2BP-CR3BP patched model accurately transition into a higher-ﬁdelity ephemeris model,

as represented in Canales et al., 2021.27

18

Figure 19. Interpolation between Moon-to-Moon Tides map (left) and Upper-

and Lower-Constraint Arrival maps (upper-right and lower-right, respectively). The

1.242 isoline is used as a reference value to evaluate both maps for a connection.

ACKNOWLEDGEMENTS

Assistance from colleagues in the Multi-Body Dynamics Research group at Purdue University is

appreciated as is the support from the Purdue University School of Aeronautics and Astronautics and

College of Engineering including access to the Rune and Barbara Eliasen Visualization Laboratory.

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