Moon-to-Moon Transfer Methodology for Multi-Moon Systems in the Coupled Spatial Circular Restricted Three-Body Problem

Abstract

Given the interest in future space missions devoted to the exploration of key moons in the Solar System and that may involve libration point orbits, an efficient design strategy for transfers between moons is introduced that leverages the dynamics in these multi-body systems. A general methodology for transfer design between the moons in any given system is developed within the context of the circular restricted three-body problem, useful regardless of the orbital planes in which the moons reside. A simplified model enables analytical constraints to determine the feasibility of a transfer between two different moons moving in the vicinity of a common planet.

Full text

AAS 20-462
MOON-TO-MOON TRANSFER METHODOLOGY FOR
MULTI-MOON SYSTEMS IN THE COUPLED
SPATIAL CIRCULAR RESTRICTED THREE-BODY PROBLEM
David Canales∗
, Kathleen C. Howell†
, and Elena Fantino‡
Given the interest in future space missions devoted to the exploration of key moons
in the Solar System and that may involve libration point orbits, an efficient design
strategy for transfers between moons is introduced that leverages the dynamics
in these multi-body systems. A general methodology for transfer design between
the moons in any given system is developed within the context of the circular
restricted three-body problem, useful regardless of the orbital planes in which the
moons reside. A simplified model enables analytical constraints to determine the
feasibility of a transfer between two different moons moving in the vicinity of a
common planet.
INTRODUCTION
The exploration of planetary moons has always played a crucial role for an improved understand-
ing of the Solar System and in the search for life beyond Earth. Past milestone missions to the
gas giants incorporated tours of the moons and, thus, satisfied multiple science objectives at dif-
ferent targets simultaneously. For example, for the Galileo spacecraft, launched in 1989, engineers
designed a 23-month tour in the Jovian system that was successfully accomplished1in the 1990s.
Another well-designed tour was implemented for the Cassini-Huygens mission, launched in 1997,
in the Saturn system,2demonstrating the advantage of tours for these types of missions in the 2000s.
From these two missions, among others, scientific questions arise to be addressed in new mission
proposals involving various worldwide space agencies. The ESA’s JUICE mission,3for example, is
proposed to study Ganymede, Callisto and Europa. The NASA missions include Europa Clipper4
for the exploration of Europa, as well as Dragonfly,5whose goal is landing a robot on Titan.
To accomplish the design of increasingly complex mission scenarios requires effective trajectories
with low propellant requirements, to support the construction of transfers between moons in a multi-
body environment. One proposed strategy to tackle this problem is leveraging the coupled circular
restricted three-body problem (coupled CR3BP) and locating connections between moons using
Poincaré sections.6–8 Generally, in such analyses, the identification of a suitable relative phase
between the moons is not considered, and many studies assume that lunar orbits are coplanar. An
alternative approach,9–11 also a coplanar analysis, propagates departure and arrival trajectories to
∗Ph.D. Student, 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 at Department of Aerospace Engineering, Khalifa University of Science and Technology, P.O. Box
127788, Abu Dhabi, United Arab Emirates; elena.fantino@ku.ac.ae
1

a certain distance from the moons, evaluates the osculating elements to represent these trajectories
as conic arcs and determines any available tangential connections, producing an optimal relative
phase between the moons as well as an analytically minimum-∆voption. However, despite having
evaluated the error of the conic approximation9in the CR3BP, the result is not yet validated with
a higher-fidelity model such as the coupled CR3BP or an ephemeris model. Of course, Lambert
arcs12 also offer the potential to identify the connection between invariant manifolds for two bodies
moving around a common body.
Numerous previous investigations also involve gravity assists in multi-moon systems; strategies
to exploit the moon gravity are examined via Tisserand graphs in the two-body problem (2BP),13, 14
the CR3BP15 or a 2BP-CR3BP patched model.16, 17 Most preliminary studies using this technique
include the assumption that the moons are in coplanar orbits. Such an approach reduces the ∆vfor
maneuvers, but the transfer time of flight (ttot) is sometimes considerably increased and the tour
design process is computationally intensive. Although all the moons can be eventually encountered,
the moon tour design is also challenging for other complex mission scenarios, e.g., leveraging libra-
tion point orbits, captures or even returning to a previously visited moon, since the gravity assists
sequence is affected.
The focus of the present investigation is an alternative methodology for transfer design between
moons for any given system; in particular, the use of dynamical structures in the three-body problem
is examined. Using an analytical formulation from the spatial 2BP, this strategy enables construc-
tion of transfers in the coupled spatial CR3BP between moons in different planes. Also, for any
transfer between moons where the analytical formulation holds valid, and for any given angle of
departure from the departure moon, a potential transfer configuration is produced. As a result, addi-
tional insights are possible when designing tours. For example, potential new options may emerge
throughout the course of the tour, such as transfers between different periodic orbits or alternatives
for captures.
The dynamical models used for this investigation are first introduced. The analytical approach is
then detailed to create a useful methodology for moon-to-moon transfers. Also, results are compared
between a coplanar assumption for the moons orbits and moon motion modeled in terms of their
actual inclined planes. A spatial application of the methodology is applied to transfers between
Titania and Oberon. Finally, some concluding remarks are offered.
DYNAMICAL MODEL
When analyzing the motion in the vicinity of a moon, the CR3BP18 is a simplified model that
offers useful insight into trajectories and the general dynamical flow for many different applications.
Designs constructed within such a framework are useful for preliminary analysis since they are
usually straightforwardly transitioned to higher-fidelity ephemeris models.
For the purposes of this investigation, the CR3BP describes the motion of a spacecraft (s/c) sub-
ject to the gravitational force of a larger primary (planet) and a smaller primary (moon), both as-
sumed to be in circular orbits about their common barycenter. In fact, most moons in the Solar
System possess very small orbital eccentricities (e); the Earth-Moon system is characterized by one
of the largest eccentricities at a value of 0.055. To model the problem, a system of differential
equations in the CR3BP is written in dimensionless form such that the characteristic distance is the
constant measure that defines the distance between the planet and the moon, and the characteristic
time is selected to guarantee a dimensionless mean motion of the primaries equal to unity. The
2

mass ratio µ=mmoon/(mmoon +mplanet )is defined as the characteristic mass parameter of the
system, with mmoon and mplanet the masses of the moon and the planet, respectively. A rotating
barycentric frame is defined 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 angular momentum vector of the system.
The planet and the moon are located at positions ¯rp= [−µ, 0,0]Tand ¯rm= [1 −µ, 0,0]T, re-
spectively. Note that overbars denote vectors whereas the subscript ’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 defined
by the following equations of motion:
¨x−2 ˙y=δU ∗
δx ¨y+ 2 ˙x=δU∗
δy ¨z=δU ∗
δz ,(1)
where dots are derivatives with respect to dimensionless time. All vectors are expressed in rotating
coordinates. Then, U∗=1−µ
rp−s/c +µ
rm−s/c +1
2( ˙x+ ˙y)represents the pseudo-potential function of
the system of differential equations, and rp−s/c and rm−s/c are the distances between the planet
and the s/c as well as the moon and the s/c, respectively. The Jacobi constant (JC) is defined as
the "energy" of the s/c in the given system via C= 2U∗−p˙x2+ ˙y2+ ˙z2. Finally, note that
motion exists in the vicinity of five equilibrium solutions in the given formulation. Such motion is
frequently categorized by different types of families of periodic orbits according to their geometry
and stability. Examples are the planar Lyapunov orbits family and the three-dimensional Halo orbits
family. These orbits are leveraged for many different types of mission scenarios. Consistently with
their stability properties, invariant manifolds that emanate from such periodic orbits often serve as
pathways in the vicinity of the moons and to locate connections between periodic orbits within the
same system.19
With one planet and two moons, each planet-moon system possesses its own characteristic quan-
tities and angular velocities, and the moons’ orbits are not located in the same plane. Given that the
CR3BP only incorporates two bodies affecting the dynamics of the s/c, it is challenging to construct
valid preliminary trajectories to travel from the vicinity of one moon to another using the CR3BP.
Hence, it is useful to leverage some simplifications. The multi-body methodology for tour design
is introduced using one sample scenario that involves a moon-to-moon transfer in the Jupiter’s sys-
tem; that is, trajectories departing from the vicinity of Ganymede and arriving in Europa’s vicinity
(see Table 1 for system’s data). Since the orbital eccentricities of the two moons are small, Jupiter-
Ganymede and Jupiter-Europa dynamics can be approximated with the CR3BP. The planes of the
moon orbits are defined by the appropriate epoch in the Ecliptic J2000 frame, but the moons move in
their respective circular orbit in their recognized orbital plane. The two models to tackle this prob-
lem are introduced below: (a) a two-body/three-body patched model (2BP-CR3BP patched model)
and (b) a coupled three-body formulation.
Table 1. Orbital data of Europa and Ganymede (SPICE,20 last accessed 08/05/2020).
Inclination Right Ascension
Semi-Major Orbital Period µ e measured from of the
Axis [day] [10−5] Ecliptic J2000 Ascending Node
[km] [degree] [degree]
Europa 6.713 ·1053.554 2.52802 0.00917 2.150 331.361
Ganymede 10.706 ·1057.158 7.80435 0.00254 2.208 340.274
3

2BP-CR3BP patched model
The 2BP-CR3BP patched model9approximates trajectories modeled in either the CR3BP or the
2BP depending on the location of the s/c in the system. Trajectories in the vicinity of a moon are
modelled with the CR3BP. When these trajectories reach a certain distance from the moon, the
motion is considered to be Keplerian with a focus at the larger primary and uniquely determined
by the osculating orbital elements: semi-major axis (a) , eccentricity (e), right ascension of the
ascending node (Ω), inclination (i), argument of periapsis (ω) and true anomaly (θ). For an example,
Figure 1(a) illustrates one unstable manifold departing a Lyapunov orbit near the L1libration point
in the Jupiter-Ganymede system. When the path reaches the sphere of influence (SoI) of the moon
(see below), the state is transformed from the rotating frame to the Ecliptic J2000 Jupiter-centered
inertial frame, and an osculating Keplerian orbit is obtained (Figure 1(b)). Similarly, trajectories
arriving at the Europa vicinity are propagated backwards in time towards the SoI, where the state is
instantaneously defined as a back-propagated Keplerian orbit in the inertial frame with Jupiter at its
focus. Thus, both Jupiter-Ganymede (J-G) and Jupiter-Europa (J-E) systems are blended from their
respective rotating frames into the Ecliptic J2000 Jupiter-centered inertial frame.
(a) Unstable manifold leaving the L1Lyapunov or-
bit in the Jupiter-Ganymede rotating frame. JC =
3.0061
(b) Departure conic after the manifold crosses
Ganymede’s SoI in the Ecliptic J2000 Jupiter-
centered inertial frame.
Figure 1. Patched 2BP-CR3BP model for the Jupiter-Ganymede system.
The transformation of a state from the rotating frame to the Ecliptic J2000 Jupiter-centered inertial
frame derives from a rotation that is easily determined with the following formulation. Assuming
that the moons move on circular orbits defined in terms of an epoch using the Ecliptic J2000 frame,
the position (¯r) and velocity ( ˙
¯r) of the s/c are obtained in the Ecliptic J2000 Jupiter-centered iner-
tial frame given Ωmoon,imoon,amoon and θmoon . Note that the moon location is evaluated using
θmoon =st +θ0, where sis the angular velocity of the moon given the period in Table 1, tis the
actual time in seconds, and θ0is the angle of the moon with respect to the ascending node line at
the time of departure (t= 0). Given the moon’s position (¯rmoon) and velocity ( ˙
¯rmoon) at a time tin
the Ecliptic J2000 frame, employ ¯
hmoon =¯rmoon×˙
¯rmoon
|¯rmoon×˙
¯rmoon|, i.e., the angular momentum of the moon
at the given time along the trajectory. The rotation matrix is represented by R= [ˆxTˆyTˆzT], such
that ˆx=¯rmoon
|¯rmoon|,ˆz=¯
hmoon
|¯
hmoon|and ˆy=ˆz׈x
|ˆz׈x|define the axes of the planet-moon rotating frame at
the given time. Note that boldface denotes a matrix. Given that the s/c rotating frame position and
velocity states (¯rrot and ˙
¯rrot , respectively) are computed relative to the barycenter, µis added to
4

Figure 2. Gravitational acceleration variation due to Jupiter and Ganymede along the ˆxaxis.
the x-state of the position vector to shift the origin to the planet, but, the velocity components are
the same after a translation. Using the basic kinematic equation, the velocity in the inertial frame,
expressed in the rotating basis, is computed as ˙
¯rin =˙
¯rrot + ¯srot ׯrrot, being ¯srot = [0 0 s]T. As a
result, the position and velocity in the inertial basis are defined by ¯r=R¯rrot and ˙
¯r=R˙
¯rin.
The SoI is defined consistently with the variation in the gravitational acceleration of the planet
and the moon along the ˆxaxis. An example for Jupiter and Ganymede of such a gravitational
influence variation is provided in Figure 2. The radius of the SoI (RS oI ) is measured from the moon
to the location where the gravitational acceleration ratio caused by the two primaries along the ˆx
axis corresponds to aSoI =amoon
aplanet = 5 ·10−4. Then, it is traced as a sphere surrounding the moon
vicinity. The ratio value aSoI = 5 ·10−4is selected as the threshold since it is a sufficiently low
moon gravitational acceleration for the motion of the s/c to be simplified to a planet-centered conic.
If such a ratio is increased, the moon perturbation increases and the approximation may not be as
valid. If the ratio is decreased, links between Jupiter-Ganymede and Jupiter-Europa trajectories
might be missed and ttot increases.
Coupled spatial CR3BP model
The coupled spatial CR3BP model considers trajectories from two different systems, each mod-
eled in the CR3BP. For example, to design a trajectory from Ganymede to Europa, Jupiter-Ganymede
and Jupiter-Europa CR3BP systems are overlapped at the common body, i.e., Jupiter. Assume that a
transfer between Ganymede and Europa is the goal. Since the fundamental planes for both systems
are different, use the following steps: CR3BP trajectories departing Ganymede are transformed from
the rotating frame to the Ecliptic J2000 Jupiter-centered inertial frame; then, the resulting trajectory
is rotated to the Jupiter-Europa rotating frame and a link between both trajectories is sought (Figure
3). In this frame, a Poincaré section at a certain angle with respect to the rotating Jupiter-Europa
ˆx-axis is used to locate connecting trajectories between Ganymede and Europa.
The transformation of a state from the Ecliptic J2000 Jupiter-centered inertial frame to the rotating
frame follows a reverse rotation. The rotation matrix is again defined by R= [ˆxTˆyTˆzT], being
ˆx=¯rmoon
|¯rmoon|,ˆz=¯
hmoon
|¯
hmoon|and ˆy=ˆz׈x
|ˆz׈x|the axes of the planet-moon rotating frame at the given
time. Given the angular velocity of the moon in the inertial frame, ¯sin =R¯sT
rot, the state of the s/c
in the rotating frame is expressed as ¯rrot =RT¯rand ˙
¯rrot =RT(˙
¯r−¯sin ׯr). Finally, to locate the
5

Figure 3. In red, unstable manifolds from L1Lyapunov orbit in the system (J Cd=
3.0057). In blue, stable manifolds from from L2Lyapunov orbit in the Jupiter-Europa
system (JCa= 3.0028). Poincaré section at 180◦; Jupiter-Europa rotating frame.
state relative to the barycenter, µis added to the xposition component.
METHODOLOGY
Consider a s/c located in an L1Lyapunov orbit in the Jupiter-Ganymede system with a Jacobi
Constant value JCd= 3.0061. Let the target destination be an L2Lyapunov orbit (JCa= 3.0024)
in the Jupiter-Europa system. Unstable manifolds depart the L1Lyapunov orbit in the J-G system
and stable manifolds arrive at the L2Lyapunov orbit in the J-E system. Determining the natural
intersection between these manifolds presents a challenge. This methodology consists of first using
the 2BP-CR3BP patched model to analytically explore promising trajectories and configurations
between two different moons. Once a promising transfer is uncovered, such a transfer serves as an
initial guess for the coupled spatial CR3BP. For the sake of comparison, the problem is explored
assuming the orbits of the moons are coplanar and, then, in different planes. Note that the strategy
applies for any moon-to-moon transfer with a common planet.
Moons on coplanar orbits
Analytical constraint for a successful transfer. Assume, for now, that the orbits of Ganymede
and Europa are in the same plane. The 2BP-CR3BP patched model is used to construct analytical
connections between trajectories in the different systems. Trajectories departing Ganymede’s vicin-
ity are propagated towards the Ganymede SoI and, when the s/c reaches it, a departure conic with
orbital elements dependent on the initial Ganymede epoch angle θ0Gan is obtained. Desirable trajec-
tories for arrival in Europa’s vicinity are also back-propagated to the Europa SoI, where an arrival
conic with orbital elements dependent on the Europa arrival epoch angle θ4Eur is produced. Recall
that the subscript ’4’ is used to represent the arrival instant. The objective is the determination of the
geometrical condition that both departure and arrival conics must possess for intersection.21 Two
confocal ellipses may have up to two point in common,9rint, as long as
rint =pd
1 + edcos θd
=pa
1 + eacos θa
,(2)
where adand aaare the semi-major axes, edand eaare the eccentricities, θdand θathe true anoma-
lies at the intersection of the departure and arrival conics, respectively, and pd=ad(1 −e2
d)and
6

pa=aa(1 −e2
a). The angle θais defined as θa=θd−∆ω, where ∆ω=ωa−ωd. Given the
trigonometric property cos(θd−∆ω) = cos θdcos ∆ω+ sin θdsin ∆ω, the following expression is
produced:
pd−pa+ (pdeacos ∆ω−paed) cos θd=−pdeasin ∆ωsin θd.(3)
Squaring both sides of Eq. (3) yields the quadratic equation k1cos2θ1+ 2k2cos θ1+k3= 0 with
the following solution:
cos θd=−k2±pk2
2−k1k3
k1
,(4)
where k1= (pdeacos ∆ω−paed)2+ (−pdeasin ∆ω)2,k2= (pd−pa)(pdeacos ∆ω−paed)and
k3= (pd−pa)2−(−pdeasin ∆ω)2. Looking into Eq. (4), only one solution is available when
k2
2−k1k3= 0. This limiting case corresponds to a tangential intersection between the two conics
(Figure 4). Trajectories that intersect tangentially frequently correspond to a theoretical minimum
∆v,9certainly in terms of the minimum energy for each given maneuver. From k2
2−k1k3= 0, an
ideal phase for the arrival moon is delivered such that a tangent configuration is guaranteed. The
appropriate arrival conic reorientation for a tangent configuration is defined:
cos ∆ω=2aaad−a2
a(1 −e2
a)−a2
d(1 −e2
d)
2aaadeaed
.(5)
For computational implementation, focus on the right side of Equation (5). The inequality constraint
2adaa(1 + edea)< b2
d+b2
ais obtained when 2aaad−a2
a(1−e2
a)−a2
d(1−e2
d)
2aaadeaed<−1, where bdand ba
are the semi-minor axis of the departure and arrival conics, respectively. This conditions implies
that the departure and arrival conics never intersect. When both sides of the equation equate, the
minimum condition for tangency is produced, which occurs between the apogee of the inner conic
and the perigee of the outer conic. Furthermore, when 2aaad−a2
a(1−e2
a)−a2
d(1−e2
d)
2aaadeaed>1, the inequality
constraint yields 2adaa(1 −edea)> b2
d+b2
a, that implies two intersections between the conics are
available. Therefore, when both sides of the equation equate, the maximum limiting geometrical
relationship between the ellipses emerges, one such that a tangent configuration occurs: an apogee-
to-apogee or perigee-to-perigee configuration results, depending on the properties of both ellipses.
Thus, the following geometrical constraints are produced:
2adaa(1 + edea)≥b2
d+b2
a≥2adaa(1 −edea),(6)
If the geometrical properties for two conics fulfill the inequality constraint represented in Eq. (6),
they can be reoriented and tangentially connected.
Arrival moon’s rephasing. Given an arrival conic that satisfies Eq. (6) and that is reoriented such
that it intersects tangentially with a departure conic, it is possible to compute the angle at which the
conic intersects the SoI: σ=ωa+θSoI . Such an angle is projected onto the arrival moon plane, and
is measured from its right ascension line in the Ecliptic J2000 frame; if the inclination of the arrival
moon’s plane is zero, it is measured from the x-axis in the Ecliptic J2000 Jupiter centered inertial
frame (Figure 4). Let δ= tan−1(ySoI
|xSoI |)be the angle between the intersection of the trajectory with
the SoI and the ˆxaxis in the rotating frame. Then, the angle of the moon in its orbit at the end of
the transfer, θ4Eur , is evaluated:
θ4Eur =σ−δ+2π∆ta
Pmoon
,(7)
where ∆tais the total time along the arrival CR3BP trajectory in the arrival moon vicinity, and
Pmoon is the period of the arrival moon in its orbit.22
7

Figure 4. Representation of the rephasing of the arrival moon for a tangent configu-
ration. All angles are measured projected upon the arrival moon plane.
(a) L1Lyapunov orbit’s unstable manifolds towards
Ganymede SoI (JCd= 3.0061). J-G rotating
frame.
(b) L2Lyapunov orbit’s stable manifolds towards
Europa SoI (J Ca= 3.0024) J-E rotating frame.
Figure 5. Departure trajectories from the Ganymede vicinity (left) and arrival tra-
jectories at the Europa vicinity (right).
Ganymede to Europa transfer. Unstable manifolds are propagated from the L1Lyapunov orbit
in the J-G system towards Ganymede’s SoI, where they transition into departure conics, similar
to the stable manifolds arriving into the L2Lyapunov orbit in the J-E system from Europa’s SoI,
where they become arrival conics in backwards time (Figure 5). Then, it is possible to determine
from the inequality constraint in Eq. (6) which trajectories intersect tangentially, as well as the total
∆vfor the maneuver at the intersection point after applying Eq. (5) (Figure 7(a)). The black line in
Figure 7(a) separates the lower boundary reflecting the inequality constraint in Eq. (6) ; i.e., outside
the colormap, all the departure conics are too large for any arrival conics to intersect tangentially.
The angle in Figure 7(a) corresponds to the location of departure/arrival on the manifold along the
periodic orbit, measured as reflected in Figure 6. The manifold trajectory leading to a tangential
∆vconnection between the moons is then constructed9(Figure 7). Since the orbits of the moons
are coplanar, the total ∆v,ttot and the transfer configuration are constant regardless of the angle of
departure θ0Gan (i.e., departure epoch in the Ganymede orbit).
Selection of the manifold corresponding to the minimum ∆vin such a simplified model en-
ables connections in the coupled CR3BP. To identify such links, the following angles are identified
8

Figure 6. Angle of departure/arrival of the manifold from/to the periodic orbit.
(a) All minimum ∆vtangent configurations. Angle
projected onto the x-y plane and measured counter-
clockwise from the ˆx-axis of the rotating frame.
(b) Minimum ∆vtrajectory in the patched model
CR3BP + 2BP in the Ecliptic J2000 Jupiter-
centered inertial frame. ∆v= 0.9433 km /s and
ttot = 9.47 days.
Figure 7. Transfer from unstable manifolds leaving from L1in the J-G vicinity
(JCd= 3.0057 ) to stable manifolds arriving to L2in the J-E vicinity (JCa= 3.0024),
assuming moons are coplanar.
from Figure 7(b): (a) the initial phase between the moons is computed measuring the location of
Ganymede from the Europa location at instant 0 for both; (b) a time-of-flight is determined for both
the unstable and stable manifolds at instant 2 (intersection between departure and arrival conics in
Figure 7(b)). A final converged connection between the moons then appears in Figure 8. Further-
more, the resulting trajectory possesses a very similar ttot and ∆vas compared to the patched conic
model. The initial guess from the simplified model is reasonably efficient as compared to analyzing
all possible scenarios with Poincaré sections. Consequently, the 2BP-CR3BP patched model offers
a good initial guess to construct a transfer in the planar coupled CR3BP problem. The importance
of Eq. (6) is significant: that is, a necessary condition that must be fulfilled to locate tangential
connections between planar trajectories linking two coupled CR3BP. Also, Eq. (6) can be leveraged
as a constraint to produce feasible transfers in the CR3BP where the motion of the s/c is mainly
governed by one primary and the trajectories are planar.
Moons on non-coplanar orbits
Analytical constraint for a successful transfer. The next level of complexity is a strategy to pro-
duce a successful transfer considering that both moon orbits are in different planes, with different
values of iand Ω. The design procedure to deliver a suitable transfer between the planes also
employs the 2BP-CR3BP patched model, given the advantage of initially exploring the problem
analytically. First, the departure instant θ0Gan , the initiating epoch for the transfer, is defined. Tra-
jectories from the CR3BP that are departing Ganymede are converted to departure conics at the
9

(a) Converged solution in the planar cou-
pled spatial CR3BP represented in the Jupiter-
Europa Rotating frame.
(b) Converged solution in the planar coupled spatial
CR3BP represented in the Ecliptic J2000 Jupiter-centered
inertial frame.
Figure 8. Converged solution in the planar coupled spatial CR3BP; ∆v=0.9456 km/s
and ttot=9.47 days.
SoI (whose orbital elements depend upon θ0Gan ), and trajectories arriving in Europa’s vicinity are
transformed into back-propagated arrival conics at its SoI (whose orbital elements depend upon the
arrival epoch, θ4Eur ). In this example, departure and arrival conics possess different values of Ω
and i. The objective is the determination of the geometrical condition necessary for both departure
and arrival conics to intersect, at some point in space, given a uniquely determined phase between
the moons. The angle θ4Eur is assumed free since the arrival moon orbit is rephased such that an
intersection between departure and arrival conics is accomplished. By applying spherical trigonom-
etry23 (Figure 9), it is possible to determine the angle of intersection for the departure and arrival
conics with respect to their node line (right ascension of the ascending node), udand uarepectively,
between the two orbital planes for the moon orbits, given the difference in right ascension of the
ascending nodes of the two conics, ∆Ω = Ωd−Ωa, and the inclinations of the two planes, idand
ia:
sin(ud) = sin(ia) sin(∆Ω)
sin(ψ)cos(ud) =
sin(ia) cos(∆Ω)
cos(id)−cos(ψ) tan(id)
sin(ψ)
sin(ua) = sin(id) sin(∆Ω)
sin(ψ)cos(ua) = cos(∆Ω) cos(ud) + sin(∆Ω) sin(ud) cos(id)
,(8)
where ψ= cos−1(cos(ia) cos(id)+sin(ia) sin(id) cos(∆Ω)).As a result, the true anomaly at which
the departure conic intersects the plane is evaluated as θdIntersection or θdInter section +π, depending
on the argument of periapsis for the departure conic, ωd:θdInter section =ud−ωd. The spatial
intersection of the departure conic occurs at:
rdIntersection =ad(1 −e2
d)
1 + edcos(θdIntersection +nπ),being n= 0,1.(9)
For a successful spatial connection between the departure and arrival conic, the following relation-
ship must hold for the arrival conic:
rdIntersection =aa(1 −e2
a)
1 + eacos(θaIntersection +nπ),being n= 0,1.(10)
10

Figure 9. Intersection between two planes using spherical trigonometry.23
As a result, the true anomaly for the intersection of the arrival conic at θaIntersection and θaInter section +
πis obtained:
cos(θaIntersection +nπ) =
pa
rdIntersection −1
ea
,being n= 0,1,(11)
where pa=aa(1 −e2
a). From Eq. (11), given that −1≤cos(θaIntersection +nπ)≤1, a useful
analytical condition is identified, one that must be fulfilled such that the arrival conic is rephased to
directly intersect with the departure conic:
aa(1 −ea)≤ad(1 −e2
d)
1 + edcos(θdIntersection +nπ)≤aa(1 + ea),being n= 0,1.(12)
The angle θdIntersection or θdI ntersection +πis generally defined by the iand Ωvalues for the depar-
ture and arrival planes as well as by ωd, which depends on θ0Gan. If the departure conic geometrical
properties, given a value of θ0Gan, meets the boundary conditions defined by the geometrical prop-
erties for the arrival conic, an intersection at θdIntersection ,θdInter section +πor both θdIntersection and
θdIntersection +πis possible. The lower boundary defines an arrival conic that is too small to connect
with the departure conic; the upper limit represents an arrival conic that is too large to link with the
departure conic. As a result, if the inequality constraint in Eq. (12) is fulfilled, the angle of inter-
section of the arrival conic θaIntersection is obtained and, since the intersection occurs at θaInter section
and θaIntersection +π,ωa=ua−θaI ntersection is obtained and with it, the intersection point between
the planes that leads to a smaller ∆vis selected. The optimal phase for the arrival moon to yield
such a configuration follows the same procedure as detailed for coplanar moon orbits.
Ganymede to Europa transfer. For purposes of comparison, the minimum ∆vtrajectory in the
planar case is introduced for this application. First, the unstable manifold trajectory is propagated
from the L1Lyapunov orbit in the J-G system towards the Ganymede SoI, where it evolves into
a departure conic, similarly to the stable manifold trajectory that arrives at the L2Lyapunov orbit
in the J-E system from the Europa SoI, where it is defined as an arrival conic. Given Eq. (12), it
is possible to complete a feasibility analysis to locate a transfer between the two specified moons
for a given departure angle from Ganymede θ0Gan (Figure 10). For every θ0Gan, if the black line
in Figures 10(a) and 10(b) is between the red and blue lines (lower and upper boundaries from Eq.
(12), respectively), the total ∆v,ttot, and the position of Europa at the initial epoch θ0Eur are noted.
Therefore, promising scenarios are available, such as the minimum ∆v(Figure 11(a)) or minimum
ttot configurations (Figure 11(b)). These solutions are transferred to the coupled spatial CR3BP by
using the necessary angles obtained in the 2BP-CR3BP patched spatial model: (a) measured from
the ascending node line for the departing moon plane, the initial location of the moon where the
trajectory starts and the intersection point between the planes (instant 0 and 2 in Figure 11, respec-
tively); (b) measured from the ascending node line for the arrival moon plane, the initial location
11

(a) Evaluation of the constraint Eq. (12). (b) Evaluation of the constraint Eq. (12).
(c) Phase of Ganymede and Europa with re-
spect to their ascending node direction at t0.
(d) Phase of Ganymede and Europa with re-
spect to their ascending node direction at t0.
(e) Transfer ∆vgiven a θ0Gan . (f) Transfer ∆vgiven a θ0Gan .
(g) Transfer ttot given a θ0Gan (h) Transfer ttot given a θ0Gan
Figure 10. Successful configurations with intersection at θdIntersection (left column)
and θdIntersection +π(right column).
12

of the moon where the trajectory starts and the intersection point between the planes (instant 0 and
2 in Figure 11, respectively). To produce a continuous trajectory in the coupled spatial CR3BP, a
multiple shooter serves as the basis for the differential corrections algorithm.19 Given that departure
trajectories are rotated onto the arrival moon plane, the problem is solved numerically by means
of a central difference approximation. Both the minimum ∆vand ttot converged solutions in the
coupled spatial CR3BP are thus produced (Figure 12).
(a) Minimum ∆vconfiguration. ∆v= 0.9448
km/s and ttot = 9.473 days.
(b) Minimum ttot configuration. ∆v= 0.9455
km/s and ttot = 9.471 days.
Figure 11. Resulting transfer trajectory using the 2BP+CR3BP patched model (Eclip-
tic J2000 Jupiter-centered inertial frame).
(a) Minimum ∆vconfiguration. ∆v= 0.9422
km/s and ttot = 9.473 days.
(b) Minimum ttot configuration. ∆v= 0.9428
km/s and ttot = 9.472 days.
Figure 12. Converged solution in the coupled spatial CR3BP (Ecliptic J2000 Jupiter-
centered inertial frame).
Alternative for spatial cases. When the given configuration does not fulfill the condition in Eq.
(12), yet the planar constraint represented by Eq. (6) is still satisfied, a successful transfer is guaran-
teed via two simple steps. First, the departure conic is propagated towards the intersection with the
arrival plane and a ∆vis applied to shift the conic to one with the same adand ed, but in the plane
13

of the arrival moon. Then, the resulting conic tangentially intersects the arrival conic with the same
∆vas if the transfer is accomplished assuming both moons are coplanar (Figure 13). Depending on
mission requirements, one of the two strategies is selected (Figure 14). For this specific case, despite
the fact that two ∆vs lead to lower costs than a single ∆vfor most initial epochs (i.e., θ0Gan), such
a behavior might not hold for other transfers or systems. Note that for this alternative, the subscript
"5" denotes the arrival instant due to the fact that another arc is incorporated.
(a) Departure to intermediate conic (yellow), and
tangential connection to arrival conic. ∆vtot =
1.0023 km/s and ttot = 15.02 days. 2BP-CR3BP
Patched Model.
(b) Transfer from departure arc to intermediate arc,
and connect tangentially to arrival arc. ∆vtot =
1.0101 km/s and ttot = 15.02 days. Coupled spa-
tial CR3BP.
Figure 13. Strategy to guarantee a transfer when Eq. (12) fails, but Eq. (6) is fulfilled.
Both figures are represented in the Ecliptic J2000 Jupiter centered inertial frame.
(a) Total ∆vdepending on θ0Gan . (b) Final ttot depending on θ0Gan .
Figure 14. Total ∆vand ttot for two strategies: directly from departure to arrival
conic (Direct Arrival Conic); from departure to an intermediate conic in the arrival
plane with tangential connection to arrival conic (Intermediate Conic).
Comparison between transfers for coplanar and non-coplanar moon orbits
It is possible to compare results between the solutions produced assuming coplanar and non-
coplanar moon orbits. This comparison is accomplished within the context of the example previ-
ously introduced: a transfer from a Lyapunov orbit in the vicinity of L1in the Jupiter-Ganymede
14

(a) Total ∆vdepending on θ0Gan for both assum-
ing moons are coplanar and in their real planes.
(b) Final ttot depending on θ0Gan for both assuming
moons are coplanar and in their real planes.
Figure 15. Comparison of the total ∆vand ttot for both assuming moons are copla-
nar and in their real planes. In the case of moons being in their real planes, successful
configurations with intersection at θdIntersection and θdI ntersection +πare included in
the same plot for the sake of comparison.
system towards an L2Lyapunov orbit in the Jupiter-Europa system via unstable and stable mani-
folds. If the transfer design approach assumes both moon orbits are coplanar, Figure 15 illustrates
that the total ∆vof the transfer as well as ttot remains constant regardless of the initial epoch θ0Gan .
Nevertheless, when the moons are incorporated in their true orbital planes, despite the fact that their
relative inclinations are small, and given that there is a difference between ΩGand ΩE, it is appar-
ent in Figure 15(a) that, depending on θ0Gan , the single ∆vvaries from a minimum value similar
to that obtained in the planar case to a maximum value of 1.75 km/s. Similarly, ttot varies from a
value in the planar case to nearly 12.25 days depending on θ0Gan (Figure 15(b)).
There exist some departure angles, θ0Gan , where a connection between the departure and arrival
trajectories does not occur, due to the fact that Eq. (12) is not satisfied. Generally, this work reflects
that the ∆vand ttot obtained assuming the moon orbits are coplanar are undervalued with respect to
the spatial model, since the results sensibly vary depending on the position of Ganymede at its plane
at departure (Figure 15). Although the coplanar analysis supplies preliminary information concern-
ing the transfers between the moons, this spatial technique supplies more accurate information to
exploit in real applications assuming a goal for a direct transfer. For a transfer involving a single
maneuver, the cost and time of a transfer between two moons are entirely dependent upon the initial
epoch, represented in this example with θ0Gan , with some angles providing no access whatsoever.
A SPATIAL APPLICATION BETWEEN TITANIA AND OBERON
Consider a s/c located in an L2northern halo orbit in the Uranus-Titania (U-T) system with a given
Jacobi Constant value JCd= 3.0035, and assume that the target location is an L1southern halo
orbit (JCa= 3.003) in the Uranus-Oberon (U-O) system (see Table 2 for system data). Unstable
manifolds depart from the L2halo orbit in the U-T system and, similarly, stable manifolds arrive into
the L1halo orbit in the U-O system. As illustrated in Figure 16, the unstable manifold trajectories
are propagated from the L2northern halo orbit in the U-T system towards the Titania SoI, where
they are evaluated as departure conics. The stable manifold trajectories associated with the L1
southern halo orbit in the U-O system, reach the Oberon SoI in backwards time, where the states are
transformed to arrival conics. Numerical algorithms to produce feasible transfers between spatial
15

orbits assuming coplanar moons are detailed by other authors.10 In this investigation, the spatial
strategy assumes that the moons are modeled in their respective true orbital planes and, thus, the
spatial transfer is derived directly. A desired unstable manifold trajectory and a stable manifold
trajectory that lead to spatial conics that fulfill Eq. (12) are selected, and the 2BP-CR3BP patched
model is used to deliver a useful initial guess to pass to the coupled spatial CR3BP.
Table 2. Orbital data of Titania and Oberon (SPICE,20 last accessed 08/05/2020).
Inclination Right Ascension
Semi-Major Orbital Period µ e measured from of the
Axis [day] [10−5] Ecliptic J2000 Ascending Node
[km] [degree] [degree]
Titania 4.363 ·1058.708 3.91675 0.00187 97.829 167.627
Oberon 5.836 ·10513.471 3.54363 0.00117 97.853 167.720
(a) Unstable manifolds propagated from L2
halo orbit with a J Cd= 3.0035 towards Ti-
tania SoI in the U-T rotating frame.
(b) Stable manifolds propagated towards L1halo
orbit with a J Ca= 3.003 from Oberon SoI in
the U-O rotating frame.
Figure 16. Departure manifolds from the Titania vicinity (left) and arrival manifolds
to the local Oberon region (right).
Given Eq. (12), it is possible to directly observe potential intersections between both manifold
trajectories given an initial departure angle from Titania with respect to its ascending node, θ0T it . If
the configuration is within the limits of such a constraint, a successful transfer occurs given an ideal
rephasing of the arrival at Oberon with respect to its ascending node, θ4Obe . Nevertheless, observe
in Figure 17 that, since the stable manifold trajectory is spatial when it crosses Oberon’s SoI, Ωa
and iavary depending on the arrival angle, θ4Obe ; of course, these are different from the values for
the Oberon orbit, ΩOand iO. Therefore, when the rephasing technique is applied using Eq. (7),
the angle σis defined in the plane of the conic and not in the plane of the arrival moon. Thus, the
departure and arrival conics no longer intersect. However, if Eq. (12) is fulfilled, an ideal rephasing
for such a transfer to occur does exist given a θ0Tit . Therefore, σis projected onto the plane of
the arrival moon, offering a sufficiently close initial guess to encounter the intersection using a
differential corrections scheme. Since θ0T it is fixed, the conic state departing the SoI is also fixed,
but not the propagation time at which it intersects the arrival conic, td. Then, an initial guess for
16

(a) iaevolution with respect to θ4Obe . (b) Ωaevolution with respect to θ4Obe .
Figure 17. Evolution of Ωaand iadepending on the arrival angle at Oberon.
the angle θ4Obe is constructed using the projected σinto Eq. (7). The arrival trajectory into the L1
southern halo orbit in the U-O system is back-propagated towards the SoI of Oberon in the CR3BP,
leading to an arrival conic whose Ωaand iadepend upon θ4Obe. The propagation time at which the
arrival conic intersects the departure conic, ta, is considered free. As a result, the objective is the
determination of a design variable vector, ¯
X, defined by θ4Obe ,tdand ta, that satisfies the constraint
vector, ¯
F(¯
X) = ¯
0, and ensures position continuity at the intersection between the departure and
arrival conic:
¯
F=


xd−xa
yd−ya
zd−za



=¯
0,¯
X=


θ4Obe
td
ta



.(13)
To solve for ¯
X, a Taylor series (truncated to first order) is expanded on ¯
F(¯
X) = ¯
0about the free-
variable initial guess, ¯
X0:¯
F(¯
X0) + DF(¯
X0)( ¯
X−¯
X0) = ¯
0. The vector ¯
X0is defined with the
values θ4Obe ,tdand taobtained from the projection of σonto the arrival moon plane. The matrix
DF is numerically computed:
DF =




∂(xd−xa)
∂θ4Obe
∂(xd−xa)
∂td
∂(xd−xa)
∂ta
∂(yd−ya)
∂θ4Obe
∂(yd−ya)
∂td
∂(yd−ya)
∂ta
∂(zd−za)
∂θ4Obe
∂(zd−za)
∂td
∂(zd−za)
∂ta




,(14)
and, since DF is square and can be inverted, the problem is solved using a Newton approach. Given
the sufficiently close initial guess, a converged solution is delivered in only a few iterations. As
a result, for every θ0T it , a similar feasibility analysis to that appearing in Figure 10 is produced.
Promising scenarios are, thus, available for direct transfers, such as a minimum-∆vconfiguration
(Figure 18), which is transferred to the coupled spatial CR3BP (Figure 19) by means of solving the
problem using a differential corrections algorithm.19
DISCUSSION AND CONCLUDING REMARKS
Trajectory design for transfers between different moons moving in the vicinity of a common
planet is a balance between diverse constraints, priorities and requirements to enable successful
missions. In this investigation, a method to determine transfers between two different moons mov-
ing around a common planet is presented within the context of the CR3BP. The analysis supports
17

Figure 18. Transfer from northern halo L2in the Uranus-Titania system to south-
ern halo L1in the Uranus-Oberon system. (Ecliptic J2000 Uranus centered inertial
frame). ∆vtot = 62.6m/s and ttot = 28.5days. 2BP-CR3BP patched model.
(a) Transfer between L2northern halo orbit of
Uranus-Titania system, to L1southern halo orbit of
Uranus-Oberon system in the Ecliptic J2000 Uranus
centered inertial frame.
(b) Transfer between an L2northern halo orbit of
Uranus-Titania system, to L2southern halo orbit
of Uranus-Oberon system in the Uranus-Oberon
rotating frame.
Figure 19. Converged solution in the coupled spatial CR3BP for a transfer between
an L2northern halo orbit of Uranus-Titania system, to L1southern halo orbit of
Uranus-Oberon system ∆v= 45.7m/s and ttot = 28.44 days.
transfers between libration point orbits near different moons. Structures that stem from the CR3BP
formulation are smoothly incorporated for moon tour design by blending conics from the 2BP with
arcs from the CR3BP. In particular, transfers between planar and spatial libration point orbits of two
different planet-moon systems are accomplished. The main advantage of this approach is that it
does not rely on the assumption that the moon orbits are coplanar, thus, determining direct solutions
with the moons in their true orbital planes. By means of incorporation of the 2BP-CR3BP patched
model, two analytical constraints are identified for both coplanar and non-coplanar moon orbits.
These analytical conditions determine the geometrical relationships between the conics and the or-
bital planes such that the epoch of the arrival moon is shifted for an intersection to occur. Given the
small eccentricities of the moon orbits and that moons are located in their true orbital planes at a cer-
tain epoch, the method yields phasing of the moons that is consistent with the real moon orbits. As
a result, these analytical expressions are leveraged to ease the design process and yield successful
transfers between CR3BP arcs in the coupled spatial CR3BP, where analytical solutions are difficult.
18

The tour design process is validated in two different multi-moon systems: transfers from Ganymede
to Europa as well as from Titania to Oberon are encountered. Both systems present entirely dif-
ferent moon orbital plane properties. As long as the spatial constraints are fulfilled, it is possible
to produce a transfer solution between planar and spatial CR3BP arcs connecting two moons. This
fact is also reflected in Figure 20(b), where a transfer between two moons with the same proper-
ties as Ganymede and Europa in Table 1 but located in very different planes is obtained. Given the
purely mathematical properties of the constraints, moon-to-moon transfers between CR3BP arcs are
designed both for outward and inward journeys (see also Figure 20(a) for an Europa to Ganymede
example). Note that the focus of this investigation is direct transfers from one moon to another;
using an intermediate conic remains an option that is likely less costly depending on the problem.
(a) Minimum ∆vconfiguration in the spatial cou-
pled CR3BP, from Europa to Ganymede. ∆v=
0.94 km/s and ttot = 12.197 days.
(b) Minimum ∆vconfiguration in the spatial cou-
pled CR3BP, considering ΩJ G = 100◦,iJG =
60◦,ΩJE = 200◦and iJ G = 20◦.∆v= 12.98
km/s and ttot = 11.83 days. This figure is uniquely
presented to justify the validity of the method.
Figure 20. Converged solutions in the spatial coupled CR3BP for other examples.
In conclusion, all the necessary steps are provided to produce a direct transfer between two dis-
tinct moons in their true orbital planes in the coupled spatial CR3BP. For any given angle of de-
parture from one moon, if Eq. (12) is fulfilled, a unique natural transfer between CR3BP arcs is
obtained with a single ∆v, given that the unique natural configuration between moons in their re-
spective planes is acquired for that given scenario. Therefore, for every departure angle, as long as
the configuration is between the limits in Eq. (12), the total ∆v,ttot, and the phase of the arrival
moon at departure are noted. Furthermore, by means of an analytical exploration of the problem,
the method provides the phase combinations between the moons that yield the most cost-efficient
results. Since the resulting spatial transfer is ultimately designed within the context of the coupled
spatial CR3BP, it may be more effective in transitioning to a higher-fidelity ephemeris model. Note
that the constraint in Eq. (12) is significant for designing trajectories not only in the coupled spatial
CR3BP, but also in the fundamental CR3BP when the motion is governed by the larger primary.
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
19

College of Engineering including access to the Rune and Barbara Eliasen Visualization Laboratory.
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20