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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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