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End-to-end trajectory concept for close exploration
of Saturn’s Inner Large Moons
E. Fantino⋆a, B. M. Burhania, R. Floresa,b, E. M. Alessic, F. Solanod, M.
Sanjurjo-Rivod
aDepartment of Aerospace Engineering, Khalifa University of Science and Technology, P.O.
Box 127788, Abu Dhabi (United Arab Emirates)
bCentre Internacional de M´etodes Num`erics en Enginyeria (CIMNE), Gran Capit`a s/n,
08034 Barcelona (Spain)
cIstituto di Matematica Applicata e Tecnologie Informatiche ”Enrico Magenes”, Consiglio
Nazionale delle Ricerche, Via Alfonso Corti 12, 20133 Milan (Italy)
dDepartment of Aerospace Engineering, Universidad Carlos III de Madrid, 28911 Legan´es,
Madrid (Spain)
Abstract
We present a trajectory concept for a small mission to the four inner large
satellites of Saturn. Leveraging the high efficiency of electric propulsion, the
concept enables orbit insertion around each of the moons, for arbitrarily long
close observation periods. The mission starts with a EVVES interplanetary seg-
ment, where a combination of multiple gravity assists and deep space low thrust
enables reduced relative arrival velocity at Saturn, followed by an unpowered
capture via a sequence of resonant flybys with Titan. The transfers between
moons use a low-thrust control law that connects unstable and stable branches
of the invariant manifolds of planar Lyapunov orbits from the circular restricted
three-body problem of each moon and Saturn. The exploration of the moons
relies on homoclinic and heteroclinic connections of the Lyapunov orbits around
the L1and L2equilibrium points. These science orbits can be extended for ar-
bitrary lengths of time with negligible propellant usage. The strategy enables
a comprehensive scientific exploration of the inner large moons, located deep
inside the gravitational well of Saturn, which is unfeasible with conventional
impulsive maneuvers due to excessive fuel consumption.
∗Corresponding author: elena.fantino@ku.ac.ae (E. Fantino)
Preprint submitted to Comm Nonlinear Sci (https://doi.org/10.1016/j.cnsns.2023.107458)July 26, 2023
arXiv:2305.17548v2 [astro-ph.EP] 24 Jul 2023
Keywords: Patched conics, Gravity assist, Circular Restricted Three-Body
Problem, Planar Lyapunov Orbits, Hyperbolic Invariant Manifolds, Low
Thrust, Optimal control, Saturn Moons
1. Introduction
According to the recently published Decadal Strategy for Planetary Science
and Astrobiology 2023-2032 [1], the open scientific questions about our planetary
system can only be addressed through the in situ exploration of the giant planets
and their moons. In particular, high emphasis has been given to new missions
to Saturn, including an Enceladus multiple flyby probe and lander and a Titan
orbiter. Due to the confirmed presence of liquid water, which likely extends
beneath the entire surface, Enceladus is considered the best candidate to host
life [2, 3]: as a matter of fact, in the plumes emanating from the ocean below
the icy terrain of the south pole of Enceladus, Cassini detected the presence of
chemical elements such as carbon, hydrogen, oxygen and nitrogen, which play
a key role in producing amino acids, the fundamental constituents of proteins
[4, 5, 6]. The Cassini mission raised new questions regarding Enceladus and
the other Inner Large Moons (ILMs) of Saturn, Mimas, Tethys and Dione. As
detailed in [7, 8], dynamical models of Mimas and Dione suggest the existence
of an ocean beneath their surface, but conclusive evidence is still missing. The
cause of the red streaks on Tethys is under debate, and so is the alleged young
origin of the moons despite their dense cratering. Other scientific open questions
regard the relative orbital dynamics, the surface composition and the thermal
and internal activity of these bodies.
Numerous follow-up exploration missions of Saturn and its moons have been
proposed, for example within the roadmaps of the 2013-2022 National Research
Council Decadal Survey [2] and ESA Science Programme [9]. NASA’s Dragonfly
mission to Titan [10] is scheduled for launch in 2026. E2T is a medium-class
solar-electric mission to Enceladus and Titan designed in response to ESA’s
M5 Cosmic Vision Call and aims at a launch opportunity in 2030 [11]. The
2
joint NASA-ESA TandEM to Titan and Enceladus [12], initially planned for
a launch in 2020, was eventually cancelled in favor of EJSM-Laplace as the
L-class outer Solar System mission candidate. More recent proposals include
SILENUS, a multi-lander and orbiter mission to Enceladus [13], and Moonraker,
an Enceladus multiple-flyby mission [14].
The design of the trajectory to tour the moons of a giant planet can be
divided into the following four phases: i) interplanetary transfer; ii) orbit inser-
tion around the planet; iii) transfer and orbit insertion around the first moon of
the system; iv) moon tour, including the science orbits around individual bodies
and the inter-moon transfers. All four phases are very demanding in terms of
fuel consumption and time of flight. For phases i) and ii), this was demonstrated
by Cassini/Huygens, the only Saturn orbiter to date. Propellant accounted for
more than 50% of the launch mass of the probe (5655 kg). Thanks to a charac-
teristic launch energy of 18 km2/s2and a sequence of planetary gravity assists
(GAs) with Earth, Venus and Jupiter, the spacecraft (S/C) reached Saturn in
6.7 years with a consumption of 1100 kg of fuel. The hyperbolic excess speed
was 5.55 km/s. The insertion into an elliptical orbit (2 ·104km ×9.3·106km)
around the planet was performed by means of a braking maneuver at the peri-
center of the arrival hyperbola, which produced a velocity variation ∆Vof 622
m/s. The capture and the subsequent pericenter raising maneuver consumed
1150 kg of propellant, with the remainder supporting the 13-year planetary tour
consisting of almost 300 orbits around Saturn with multiple flybys of Titan and
other major moons [15]. In summary, a large amount of propellant was burnt
to decelerate the S/C so that it could be captured by the planet’s gravity upon
arrival from interplanetary space. Inserting a S/C into orbit around one or more
moons deeply immersed in the gravity well of the planet, i.e., phases iii) and
iv) above, is even more challenging from a dynamical point of view. This en-
deavour has never been attempted. However, answering the scientific questions
listed previously requires extended observations of these bodies, hence the need
for probes orbiting them.
The first concept study of a moon tour has been the Petit Grand Tour
3
(PGT) of the Galilean moons [16, 17], where dynamical systems methods are
employed to design transfers between libration point orbits (LPOs) of the Cir-
cular Restricted Three-Body Problems (CR3BPs) formed by Jupiter and indi-
vidual moons. Coupling distinct CR3BPs allows to patch trajectories belonging
to the two systems. The connections are sought between unstable and stable
hyperbolic invariant manifolds (HIMs) of periodic orbits around the collinear
libration points [18, 19]. A tour of a system of moons can be designed with
the resonance hopping technique [20, 21, 22], which allows to patch consecutive
orbits in mean motion resonance with a given moon. The periodic perturbation
caused by the moon can be exploited to change the features of the next encounter
or to redirect the S/C towards another target. This approach can be enhanced
by applying small maneuvers in proximity of the GAs [23, 24]. The resonance
hopping concept can be envisaged also within the CR3BP [25, 26, 27]: by means
of the Tisserand-Poincar´e graph, it is possible to link different CR3BPs and iden-
tify families of periodic orbits leading to a resonant hopping tour. The case of
Saturn was analysed by [28, 29, 21, 22]. The use LPOs in this context was pro-
posed by [28] based on the observational proof of the existence of Trojans in the
Saturn-Dione and Saturn-Tethys systems. Lantoine and Russell [30] designed
a resonance hopping tour in the Jovian system by connecting resonant orbits
through HIMs of halo orbits of planet-moon CR3BPs. A different approach
searches for stable orbits around the moon. The challenge is to model and
properly exploit the gravitational field of the object to identify suitable frozen
orbits. Along this line, Russell and Brinckerhof [31] proposed mid-inclination,
high-altitude, stable, eccentric frozen orbits, obtained by doubly-averaged equa-
tions of motion of the third-body effect. They considered the Jovian moons as a
test case, but a similar analysis can be applied to the Saturn system, as shown
by [32].
The main objective of this contribution is to demonstrate a complete tra-
jectory concept from Earth departure to a tour of the ILMs with minimum fuel
consumption. The design methodology takes into account the limitations of
power and propulsion technologies in a Saturn mission, the cost of launching
4
into deep space and the expected scientific return of a tour of the major moons
of Saturn. The study makes use of state-of-the-art methodologies, including
patched conics, GAs, optimal control and dynamical systems tools. The aim
is to minimize cost (∆V) and mass while ensuring a mission timeline of an ac-
ceptable duration. The propulsion system selected is electrical, and the power
is supplied by radioisotope thermoelectric generators. As a result, the onboard
resources are extremely scarce, which limits the available thrust. For this rea-
son, the gravitational field of planets and moons is utilized to a large extent so
as to minimize the use of the thruster and limit the time of flight. The transfer
strategy is carefully planned and leverages the natural dynamics of the involved
systems: the interplanetary trajectory and the orbit insertion at Saturn make
use of GAs, whereas the tour of the ILMs is based on low-energy (LE) orbits
associated with the L1and L2equilibrium points of the CR3BPs with Saturn
and individual moons as primaries.
The main element of originality of the work resides in the unprecedented con-
cept of achieving orbit around the four moons of Saturn, using only low-thrust
(LT) propulsion and gravitational assistance. Preliminary, partial versions of
this work can be found in [33, 34, 35]. Here, we present a complete plan includ-
ing re-designed interplanetary trajectory with global optimization techniques
and revised Saturn Orbit Insertion (SOI) and transfer to Dione, the first moon
of the tour. In addition, we study the observational performance of the science
orbits around the four targets.
The paper is organised as follows: Sect. 2 defines the dynamical models
adopted in the study, Sect. 3 illustrates the design of the interplanetary tra-
jectory, Sect. 4 focuses on the gravity-assisted orbit insertion at Saturn, while
Sect. 5 is devoted to the transfer to the ILMs and the design of the tour of
Dione, Tethys, Enceladus and Mimas. A discussion of the results can be found
in Sect. 6, while the strengths and limitations of the investigation as well as an
anticipation of future work are the topics of Sect. 7. The following abbreviations
are often used in tables and figures: S = Saturn, Ti = Titan, Di = Dione, Te
= Tethys, En = Enceladus, Mi = Mimas.
5
2. Dynamical models
2.1. Interplanetary trajectory
The interplanetary portion of the trajectory is a gravity-assisted itinerary
alternating ballistic and LT arcs between planetary encounters. The dynamical
model is the two-body problem (2BP) with the patched conics approximation.
The GAs are with Venus and Earth. They are unpowered and designed as-
suming zero radius for the planetary spheres of influence (ZRSI); hence, the
pericenter altitude of the flyby hyperbolas is a free parameter. Planetary orbits
are assumed coplanar, and the trajectory of the S/C is also 2D. The positions
and velocities of the planets are obtained from JPL NAIF-SPICE kernels [36]
in the Sun-centered International Celestial Reference Frame (ICRF) and are
projected on the xy-plane as follows:
•the projection of the position vector r= (rx,ry,rz) preserves the helio-
centric distance, whereas the polar angle from the xaxis is computed as
the 2-argument arctangent of ryand rx;
•the magnitude of the 2D-projected velocity vector is equal to the magni-
tude of v= (vx,vy,vz), and the flight path angle γon the ecliptic plane
is equal to the corresponding quantity in 3D: tan γ=|r×v|/(r·v).
For the propelled phases of the transfer, a constant thrust magnitude Tis as-
sumed. Hence, the dynamical equations for the S/C in the presence of thrust
are:
˙
r=v
˙
v=−GM
r3r+T
mu
˙m=−T
g0Isp
(1)
GM being the gravitational parameter of the Sun, mthe mass of the S/C, Isp
the specific impulse of the thruster (assumed constant), uthe instantaneous
direction of thrust and g0= 9.81 m/s2. The relevant physical data for the Sun,
Venus, Earth and Saturn are listed in Table 1. A minimum altitude of 200 km
is imposed for the GAs with Venus and Earth.
6
Table 1: Relevant physical data for the Sun and the three planets involved in the study:
gravitational parameter GM and mean radius R.
Body GM R
(105km3/s2) (km)
Sun 13271 -
Venus 3.2486 6051.8
Earth 3.9850 6371.0
Saturn 379.22 60268
2.2. Saturn orbit insertion
The encounter of the S/C with Saturn occurs at the point of intersection
between the respective heliocentric trajectories. The hyperbolic excess velocity
v∞of the S/C relative to Saturn is computed as the difference between the
velocities of the S/C and Saturn. In this way, the pericenter radius of the
planetocentric hyperbola is a free parameter. The orbit insertion and the descent
towards the ILMs are carried out through a sequence of unpowered GAs with
Titan followed by a propelled arc. The GAs are modelled with the patched
conics approximation assuming ZRSI for this moon. As a result, the pericenter
altitude of the Titan-centered hyperbolas is a free parameter.
2.3. Descent towards the ILMs and moon tour
The descent from Titan to the ILMs is a propelled transfer to the vicinity
of Dione with suitable conditions to start the tour. Both the transfer to Dione
and the tour are designed in 2D in the equatorial plane of Saturn. The tour
alternates inter-moon propelled transfers and science orbits computed in the four
Saturn-moon CR3BPs and linked at suitably-defined Poincar´e sections where
the three-body states of the S/C are used to determine osculating 2BP orbits
relative to Saturn (see also [37]). In the descent to Dione and in the inter-moon
transfers, the motion of the S/C is determined by Eq. 1 with GM equal to the
gravitational parameter of Saturn and r,vrepresenting the state of the S/C in
the Saturn-centered inertial reference frame. In the science phases, the S/C is
7
subject to the gravitational attraction of Saturn (mass m1) and a moon (mass
m2), and the two primaries move in circular orbits around their centre of mass.
The equations of motion of the S/C are written with respect to the barycentric
synodic reference frame of the primaries, with these two bodies on the xaxis
[38]. Using the gravitational constant G, the sum m1+m2of the masses and
the distance dbetween the primaries as reference magnitudes, the mean motion
of the orbits of the primaries takes unitary value and their orbital period equals
2π. The larger (Saturn) and the smaller (moon) bodies are at (µ,0,0) and (µ-
1,0,0), respectively, µbeing the mass ratio m2/(m1+m2) of the system. Table 2
lists the main physical and dynamical parameters of the ILMs and Titan and
the relevant data of the four CR3BPs.
Table 2: Main physical and orbital parameters of the four ILMs and Titan (mean physical
radii R, orbital periods P, orbital radii d), and relevant data of the four Saturn-moon CR3BPs
(mass ratios µand x-coordinates of L1and L2in the moon-centered synodic reference frames).
Moon R P d µ x(L1)x(L2)
(km) (day) (km) (10−6) (km) (km)
Mi 198 0.9424 186000 0.06599 521 -521
En 252 1.370 238000 0.18993 947 -950
Te 533 1.888 295000 1.08660 2097 -2107
Di 561 2.737 377000 1.92799 3245 -3262
Ti 2575 15.95 1222000 - - -
3. Design and optimization of the interplanetary trajectory
To enable practical orbit insertion around the inner moon of Saturn using
only low thrust, our mission concept relies on an interplanetary trajectory with a
low hyperbolic excess speed at arrival. Fantino et al. [39] showed the possibility
of having an interplanetary trajectory with an arrival hyperbolic excess speed of
1 km/s. As explained in Sect. 6, this low relative velocity simplifies unpowered
capture via a Titan flyby. In [39], the mission design was based on LT electric
8
Table 3: Thrust T, input power Pin and specific impulse Isp of three settings of the PPS X00
thruster studied [41].
Setting T Pin Isp
(mN) (W) (s)
1 36 640 1600
2 40 650 1450
3 50 850 1400
propulsion and a single unpowered GA. The parameters of the propulsion system
corresponded to the NEXT ion engine [40] with 25 mN of thrust, 1400 s of
specific impulse and 600 W of power consumption. The initial mass of the S/C
was 1500 kg with a characteristic launch energy of 67.25 km2/s2. The two legs of
the itinerary, i.e., Earth-Jupiter and Jupiter-Saturn, were optimized separately
and without considering phasing constraints for the planets. In the first leg, the
thruster was continuously fired in the direction of motion. In the second leg, a
locally-optimal guidance law was used to determine the direction of thrust that
allows to reach Saturn with a hyperbolic excess speed of 1 km/s. The transfer
duration to Saturn was 13 years, consuming 367 kg of propellant (propellant
mass fraction = 0.367).
In the present work, the interplanetary trajectory design is refined to obtain a
more realistic solution. The launch characteristic energy is reduced and phasing
constraints for the GAs and arrival at Saturn are included, considering the
ephemeris of the planets. Additionally, to accommodate the propellant required
for transfers inside the Saturn system, the launch mass is increased to 1500 kg,
and the parameters of the propulsion system are changed to those of a PPS X00
Hall thruster [41]. This propulsion system generates acceptable thrust levels (up
to 60 mN) with limited power consumption (below 1000 W). Three points from
the performance envelope, with input power comparable to the value assumed
in [39], have been considered for the trajectory analysis (Table 3).
The optimization of the trajectory is performed using an automated hybrid
9
optimal control method. This is a two-stage approach [42]: 1) in the first step,
a multi-objective heuristic algorithm explores the global solution space using a
surrogate dynamical model, which assumes a predefined shape for the thrust
arcs (generalized logarithmic spirals [43]). The first step determines the integer
optimization variables (i.e., the number and sequence of GAs); 2) in the sec-
ond step, a single-objective direct collocation method includes a full dynamical
model and complex constraints for obtaining a higher-fidelity solution; a result
from the previous step is used as an initial guess.
The heuristic global search (step 1) has been conducted assuming a maxi-
mum thrust level of 50 mN and a specific impulse of 1400 s (setting #3 in Table
3). A maximum interplanetary transfer duration of 15 years is assumed, together
with a minimum characteristic launch energy of 16 km2/s2. The minimum char-
acteristic launch energy that met the 15-year constraint was 27 km2/s2, which
is the value used for all the results presented henceforth. Itineraries involving
intermediate GAs with Earth, Venus and Jupiter were considered. However,
because the sequences including Jupiter required excessive transfer times, they
are not presented here. A GA with Jupiter is usually desirable to reduce trans-
fer times. It becomes much less attractive when the 1 km/s relative velocity
constraint on arrival is introduced. A strong boost from Jupiter tends to in-
crease the arrival excess speed, requiring in turn more propellant and time for
the interplanetary transfer (i.e., the propulsion system needs more time to bleed
mechanical energy from the spacecraft before arrival).
The heuristic search is performed using a multi-objective genetic algorithm
(NSGA-II [44]) to minimize the total transfer time and the propellant mass frac-
tion. The algorithm has been run with populations of 100 individuals, which
evolve over 500 generations. The obtained optimal trajectories correspond to
two itineraries: Earth-Venus-Earth-Earth-Saturn (EVEES) and Earth-Venus-
Venus-Earth-Saturn (EVVES). They are depicted in the Pareto front of Fig. 1.
The transfer times are substantially longer (above 12 years) than for conven-
tional mission concepts. This is a direct consequence of the requirement for a
small relative arrival velocity. It results in a slow rate of approach to Saturn,
10
Figure 1: Propellant mass fraction vs. total transfer time for the optimal solutions of the
global heuristic search.
automatically making the last interplanetary segment very long.
The EVVES solutions are characterized by lower propellant consumption
and longer transfer times than the EVEES trajectories. When used as initial
guesses for the second optimization stage, EVEES trajectories yield converged
solutions only for the highest thrust setting in Table 3 (50 mN). EVVES tra-
jectories, on the other hand, can be used for all settings (36 to 50 mN). The
solution circled in black in Fig. 1, corresponding to a transfer time of 14.06 years
and a propellant mass fraction of 0.176, strikes a good compromise between the
two objectives and has been further optimized with the direct method (step 2).
Applying different relative weights (w) to the transfer time tfand the final mass
mfin the objective function J
J=−mf+w·tf,(2)
and decreasing the maximum thrust yields the results reported in Table 4: for
each of the seven solutions, the table lists the throttle setting, transfer time
tf, total impulse I=RtT
0|aT|dτ(with aTdenoting the thrust acceleration),
duration tTof the thrust interval, propellant mass fraction pmf, launch date
TLand arrival date TA. Solution #4 (highlighted in boldface) is the best com-
11
promise between propellant consumption and transfer time. Figure 2 illustrates
the complete trajectory and an expanded view of its inner portion, while Fig. 3
highlights the propelled arcs (thick solid lines) and ballistic segments (dotted
lines). The thrust magnitude and thrust angle histories are plotted in Fig. 4,
where an angle of zero corresponds to thrusting in the circumferential direction.
Table 4: Optimal solutions obtained from the selected trajectory in Fig. 1 through the direct
collocation method for different maximum thrust levels and cost function weight parameter w
.
Sol. Thrust w tfI tTpmf TLTA
# setting (kg/year) (year) (km/s) (year) (yyyy/mm/dd) (yyyy/mm/dd)
1 1 0 14.8 5.83 6.50 0.311 2023/02/09 2037/12/22
2 1 188 11.8 7.59 8.10 0.383 2023/02/02 2034/11/20
3 1 157 12.0 7.86 8.34 0.394 2025/01/01 2036/12/25
4 1 118 12.3 6.16 6.81 0.324 2028/01/21 2040/05/24
5 2 0 14.8 6.18 6.02 0.352 2023/01/01 2037/10/23
6 2 188 12.0 6.57 6.37 0.370 2028/01/20 2040/01/02
7 3 0 14.2 5.60 4.43 0.335 2023/02/16 2037/04/27
Figure 2: The optimal EVVES trajectory (left) and expanded view of its inner portion (right).
4. Saturn orbit insertion
The low hyperbolic excess speed v∞of 1 km/s enables unpowered capture
via a GA with Titan. This is important, because the concept of the mission
12
Figure 3: Propelled (thick lines) and ballistic (dotted lines) arcs of the optimal trajectory.
Figure 4: Thrust magnitude (left) and thrust angle (right) histories for the optimal trajectory.
(low-thrust propulsion only) does not allow for impulsive maneuvers. While it
is possible to achieve ballistic SOI with substantially higher arrival velocities,
the very long period of the post-GA orbit makes this option impractical [39].
On the other hand, lower excess speeds offer little advantage for orbit insertion
—the Saturn inbound hyperbola is very similar to a parabola— but result in
longer interplanetary transfer times with increased propellant consumption. A
relative velocity of 1 km/s offers a reasonable trade-off, and has been retained
for all the calculations. Obviously, it would be possible to include v∞as a free
optimization parameter in order to improve the results. However, to demon-
strate the mission concept and keep the discussion as simple as possible, this
refinement was deemed unnecessary.
13
The GA with Titan is performed in the inbound leg of the Saturn-centered
arrival hyperbola. Assigning a pericenter altitude of 1295 km (the lowest height
reached by Cassini above this moon was 1000 km) results in an elliptical orbit
with a period of approximately 80 days or 5 orbits of Titan, allowing the S/C
to perform another GA with this moon. A pericenter altitude of 3775 km in
this second encounter leads to a resonant orbit with twice the period of Titan.
If the third close encounter with the moon occurs at an altitude of 2258 km,
the post-GA orbit is in a 1:1 resonance state with Titan. A passage 2690 km
above the surface in the fourth GA places the pericenter of the post-GA orbit
at the radius of Dione. Figure 5 depicts the transfer from the initial hyperbola
Figure 5: Gravitational capture at Saturn and resonant GAs with Titan: arrival hyperbola
(magenta) and post-GA orbits (5:1 in blue, 2:1 in orange, 1:1 in green, orbit with pericenter
at the radius of Dione in cyan).
through the four GAs to the final orbit. The notation m:nmeans that morbital
periods of the S/C equal norbital periods of Titan. The pericenter altitude hπ
of the four GAs and the characteristics of the post-GA orbits (semimajor axes
a+, eccentricities e+and periods P+) are reported in Table 5. Note that the
resonant state causes the four Titan-centered hyperbolas to have the same v∞
of 2.962 km/s.
14
Table 5: Pericenter altitudes hπof the four GAs with Titan and semimajor axes a+, eccen-
tricities e+and periods P+of the post-GA orbits.
GA# hπa+e+P+
(km) (106km) - (day)
1 1295 3.5739 0.7195 79.79
2 3775 1.9399 0.5602 31.91
3 2258 1.2220 0.5126 15.96
4 2690 0.9298 0.5943 10.59
5. Descent to Dione and moon tour
The tour alternates LE orbits in the vicinity of the moons (Sect. 5.1) and
LT trajectories in the inter-moon transfers (Sect. 5.2), proceeding from the
outermost (Dione) to the innermost (Mimas) body. The concept of the tour is
sketched in Fig. 6. The descent from Titan to Dione has been designed with
the same technique as the inter-moon transfers. For this reason, it is described
in Sect. 5.2. The descent to Dione after the resonant flybys over Titan could be
shortened using GAs with Rhea. This possibility has not been considered for
the sake of brevity and simplicity. Furthermore, due to the small mass of Rhea,
the effect is limited compared with the Titan GAs. For a proof of concept,
neglecting the influence of Rhea gives a conservative estimate of mission length
and propellant budget, so it is acceptable. It should be included, however, in a
detailed mission design.
5.1. Science orbits
Figure 7 illustrates families of planar Lyapunov orbits (PLOs) around L1and
L2for the four CR3BPs. Each set contains 25 orbits equally spaced in Jacobi
constant CJ, with the orbits around L1and L2having the same CJvalue.
Individual orbits in the set are identified by an index running from 1 to 25.
Table 6 summarizes the characteristics of the families: minimum and maximum
CJ(CJmin ,CJ max), minimum and maximum period (Pmin ,Pmax), minimum
15
Figure 6: The moon tour: LE transfers as science orbits in the vicinity of the moons and LT
Saturn-centered trajectories to move between moons.
and maximum yamplitude (∆ymin, ∆ymax), and the orbit index corresponding
to the various extrema.
Figure 7: Families of PLOs around L1and L2for the four CR3BPs in the respective moon-
centered synodic reference frames. CoI is the circle of influence and is defined in Sect. 5.2.
16
Table 6: Main characteristics of the families of PLOs at L1and L2in the four CR3BPs
considered: minimum and maximum CJ(CJmin ,CJ max), minimum and maximum period
(Pmin,Pmax ), minimum and maximum yamplitude (∆ymin, ∆ymax ) and the orbit index
(last row) for which the extrema occur.
CR3BP LiCJmin CJ max Pmin Pmax ∆ymin ∆ymax
- - (day) (day) (km) (km)
SMi L13.000032 3.000068 0.457 0.548 288 1373
SMi L23.000032 3.000068 0.458 0.549 281 1375
SEn L13.000036 3.000138 0.664 0.923 568 3273
SEn L23.000036 3.000138 0.668 0.920 557 3275
STe L13.000114 3.000440 0.918 1.269 1240 7279
STe L23.000114 3.000440 0.922 1.272 1190 7288
SDi L13.000097 3.000640 1.324 2.038 2030 13290
SDi L23.000097 3.000640 1.339 2.043 2080 13300
Orbit index 25 1 1 25 1 25
The stable and unstable HIMs of the PLOs have been computed and prop-
agated using standard methods, i.e., an initial state is generated by applying a
small perturbation in the direction of the stable and unstable eigenvectors of the
monodromy matrix of the PLO after appropriate time transformation through
the state transition matrix (see e.g., [45, 46]). Each PLO is discretized with 100
points, each of which corresponds to an invariant manifold trajectory. Then, the
branches that develop towards the moon have been used to construct hetero-
clinic connections between PLOs with the same value of CJat the two equilibria
[47, 48, 49, 50]. The stable and unstable HIMs also serve to construct homo-
clinic connections, i.e., trajectories that depart and approach the same PLO (see
[48, 51]). By construction, for both types of transfers the cost to leave and ap-
proach a PLO is negligible. The stable and unstable segments of the transfer are
connected at a suitably defined Poincar´e section, i.e., the y-axis and the x-axis
of the moon-centered synodic reference frame, respectively, for heteroclinic and
homoclinic transfers. For each CJ, the transfer is constructed with the arcs for
which the magnitude of the velocity difference at the Poincar´e section is lowest.
This minimum impulse is at the level of 1 m/s or less in all cases. Examples of
science orbits of the two types around the different moons are depicted in Figs. 8
17
and 9, where the blue and red colors denote stable and unstable HIMs trajec-
tories, respectively. All the computed trajectories satisfy a minimum-altitude
constraint of 20 km. All the heteroclinic transfers are reversible, i.e., symmetric
connections exist from L2to L1. Moreover, the autonomous character of the
CR3BP allows to choose the departure time from a PLO arbitrarily. Hence,
the same heteroclinic/homoclinic transfer can be used multiple times to explore
the same moon. The PLOs can be used as parking orbits between consecutive
flights and as gateways to depart the vicinity of a moon and approach the next
target in the tour. Additionaly, LT transfers between PLOs of different energy
are also possible, as shown for example in [52, 53, 54].
The time history of altitude and the time coverage of the surface for the
heteroclinics and homoclinics of Figs. 8 and 9 are illustrated in Figs. 10 and
11, respectively. Here, time coverage of a surface element (the discretization
employed is 1◦×1◦) is the accumulated time during which the S/C is above the
local horizon. Note that, due to the condition of tidal lock, the moons do not spin
in the respective synodic reference frames. In Figs. 8 and 9, the prime meridian
is aligned with the positive xaxis of the moon-centered synodic reference frame.
Depending on the altitude and shape of the trajectory, the time coverage of many
areas can reach several tens of hours over individual heteroclinic or homoclinic
transfers. In many cases, the maximum visible latitude reaches 80◦, meaning
that these trajectories allow exploration of a substantial part of the polar regions
of the moons.
5.2. Descent to Dione and inter-moon transfers
Following [37], the outward branches of the stable and unstable HIMs (i.e.,
those that unfold away from the moon, see Fig. 12) are propagated until they
intersect a moon-centered circle, called circle of influence (CoI), whose radius
rCoI equals that of the Laplace sphere of influence [55] for the given moon scaled
by an ad hoc factor f≥1:
rCoI =f d m2
m12/5
.(3)
18
Figure 8: Examples of L2-to-L1heteroclinic connections in the four CR3BPs: stable and
unstable HIMs (left) and connecting arcs (right).
19
Figure 9: Examples of homoclinic connections in the Saturn-Dione and Saturn-Mimas
CR3BPs, respectively around L1and L2.
fmust be sufficiently large to ensure that the CoI encircles all the PLOs of
the two families and intersects the outward branches of the HIMs as close to
orthogonally as possible (Fig. 12). In this study, fhas been set equal to 4.5
for all the CR3BPs. Hence, the radii of the four CoIs are 8792, 5457, 2194 and
1121 km for Dione, Tethys, Enceladus and Mimas, respectively (see Fig. 7). The
state vectors of the HIMs at the CoI are then represented in the Saturn-centered
ICRF and used to compute osculating two-body orbits with focus at Saturn
(Fig. 13). Neglecting the gravitational attraction of the moon outside the CoI
turns the design of a transfer between moons in adjacent orbits into the search
for intersections between coplanar, confocal ellipses, as shown in [37] for the case
of the Galilean moons. The velocity difference between the connecting arcs at
their intersection point(s) can be compensated for by an impulsive maneuver1.
In the planar approximation, the orbital elements of the osculating ellipses
are the semimajor axis a, the eccentricity eand the longitude of the pericenter
ω. In the case of the ILMs, the osculating ellipses emanating from adjacent
moons do not intersect. Their eccentricities are very small, while the inter-moon
1[37] showed that in the case of the Galilean moons, disregarding the gravitational attrac-
tion of the moon outside the CoI introduces an error of only 0.6% in the magnitude of the
impulse at the point of intersection.
20
Figure 11: Height above the moon vs. time (left) and surface coverage (right) for the two
homoclinic connections of Fig. 9.
Figure 12: Intersections between the outward branches of the stable (left) and unstable (right)
HIMs of PLOs around L1and L2of a Saturn-moon CR3BP.
of the orbit index, with different colors for stable and unstable trajectories and
black dashed lines representing the orbital radii of the four moons. In the
case of Mimas, only solutions corresponding to L2are shown because this is the
22
Figure 13: Osculating Keplerian Saturn-centered orbits obtained from the states of the un-
stable HIM trajectories of Fig. 12 at the CoI of the given moon.
innermost moon in the tour and no transfers departing the vicinity of its L1point
are considered. For the other three moons in the itinerary, i.e., Dione, Tethys
and Enceladus, separate sets of values appear for orbits emanating from L1and
L2. The lack of overlap between ranges associated with distinct moons proves
that direct impulsive transfers between ILMs are not possible. Figures 15 and
16 illustrate the orbital semimajor axes and eccentricities of all the osculating
ellipses.
Figure 14: Pericenter and apocenter radii of the osculating Keplerian orbits corresponding
to stable and unstable HIM trajectories emanating from PLOs around L1/L2of the four
CR3BPs.
23
Figure 15: Semimajor axes of the osculating Keplerian orbits corresponding to stable and
unstable HIM trajectories emanating from PLOs around L1/L2of the four CR3BPs. The
orbit approaching Dione through L2with the largest semimajor axis and indicated with an
arrow is the target of the descent from Titan to Dione.
The approach adopted in this work to design inter-moon connections is based
on LT transfers. The concept is sketched in Fig. 17 for a trajectory from a PLO
around the L1point of Dione to a PLO around the L2point of Tethys. Moon-
to-moon transfers in the tour always originate in the vicinity of the L1point of
the outer moon and approach the L2point of the inner one. The direction of the
transfer determines the stability character of the HIMs at each end, i.e., unstable
at the outer moon, stable at the inner moon. The propelled arc starts and ends
at the CoIs of the departure and arrival moons. In the case of the descent from
Titan to Dione, the LT transfer aims to connect the last post-GA orbit (see
Table 5) and the osculating ellipses corresponding to the stable manifolds of
PLOs around L2in the Saturn-Dione CR3BP. In all cases, uninterrupted thrust
at 36 mN (specific impulse of 1600 s) is applied in a direction determined locally
by a guidance law that maximizes the instantaneous rate of reduction of an error
function ℑexpressed in terms of the osculating orbital elements. The strategy
aims at modifying the departure elements (a,e) and making them coincide with
24
Figure 16: Eccentricities of the osculating Keplerian orbits corresponding to stable and un-
stable HIM trajectories emanating from PLOs around L1/L2of the four CR3BPs.
Figure 17: The inter-moon transfer (here from Dione to Tethys) is an LT trajectory between
osculating orbits originating from the PLOs of the respective CR3BPs.
25
those (¯a, ¯e) of the arrival orbit. The form chosen for the error function ℑis
ℑ= (a−¯a)2+ (ae −¯ae)2,(4)
in which the eccentricity is multiplied be the semimajor axis to keep the ex-
pression dimensionally homogeneous and avoid large differences in the order of
magnitude of the two components of the error when natural units are not used.
The typical time scale of this LT transfer is at least one order of magnitude
longer the than orbital periods of the moons. Therefore, the phasing require-
ments can be ignored, since the departure time can be adjusted suitably without
a significant effect on the transfer duration. In fact, it is even possible to use
an arbitrary departure time and correct the phase a posteriori, with a slight re-
duction of the thrust magnitude to introduce the appropriate delay. With this
feature in mind, the error function is assumed to depend only on the semimajor
axis and eccentricity, i.e., ℑ(a, e). This is not a requirement of the method, but
it helps in keeping the expressions simple. What follows is a direct application
of the classical gradient descent optimization technique [56]. In the context of
modern guidance algorithms, it can be considered a particular case of the general
Proximity Quotient guidance law (Q-Law) [57, 58, 59]. Details of the algorithm
can be found in [39]. Note that since the thrust is constant, the time-optimal
transfers are also the most efficient in terms of propellant consumption.
For the descent from Dione to Titan, there is one departure orbit and 100 ×
25 possible sets of arrival orbital elements. The optimal (i.e., minimum-∆V)
trajectory connects to the PLO with index 25 of Saturn-Dione CR3BP. The
arrival osculating ellipse has the largest semimajor axis in the set (see Fig. 15)
and, consequently, the smallest energy difference with respect to the last post-
GA orbit. This highlights the importance of the energy difference in the cost of
the transfer. However, energy is not the only factor to considere, eccentricity
also plays a role. This is demonstrated by the inter-moon transfers (see next
paragraph) where the most efficient solution is not always between orbits with
the minimum energy gap. For the Dione to Titan transfer, the S/C performs
290 revolutions around Saturn. The time of flight is 1574 days, the propellant
26
consumption is 312 kg and the velocity variation is 5770 m/s. The mass of the
S/C upon arrival at Dione is 702 kg. As indicated at the beginning of Sect. 5, the
transfer time could be improved performing GAs with Rhea during the descent.
Figure 18 shows the variation of semimajor axis (from 9.3·105km to 4.0·105
km), eccentricity (from 0.5943 to 0.0535), pericenter and apocenter radii over
the transfer. Figure 19 illustrates the thrust angle (measured clockwise from
the circumferential direction) over the first three revolutions.
Figure 18: Variation of semimajor axis and eccentricity (left), pericenter and apocenter radius
(right) over the optimal LT transfer from Titan to Dione.
Figure 19: Thrust angle history over the first three revolutions around Saturn during the LT
transfer from Titan to Dione.
For the inter-moon transfers, 100 ×25 sets of initial conditions are available
at each end, resulting in 6.25 ·106candidate trajectories. The contour maps
27
of Fig. 20 depict the minimum velocity variation (∆V) over all the trajectories
connecting a given pair of PLOs. Due to the large amount of possible combi-
nations of departure and arrival conditions, only the optimal solutions for each
orbit pair have been represented (the indices of the optimal arrival and depar-
ture PLOs can be found in Table 7). As expected, the lowest cost transfers are
between PLOs whose osculating ellipses have reduced energy gaps. However, as
stated before, eccentricity is also a factor. The optimal departure tra jectories
do not have the largest yamplitude (lowest CJ, corresponding to index 25).
Therefore, they do not minimize the difference of semimajor axes. However, the
change in ∆Vbetween the optimal solution and the transfer between ellipses
of minimal energy gap in small (tens of meters per second, at most). Thus, the
minimal semimajor axis difference criterion could be used to obtain a prelimi-
nary estimate of the total impulse, without having to explore all the trajectory
combinations.
Table 7 reports the performance characteristics of the optimal solution for
each segment of the itinerary (columns 2-4). For the sake of comparison, the
LT circle-to-circle transfers between moons (orbital radii diand dj) have been
computed assuming tangential thrust and Edelbaum’s analytical expression for
the velocity variation [60],
∆V=rGM
di
−sGM
dj
,(5)
with GM the gravitational parameter of Saturn. The corresponding perfor-
mance is included in Table 7 (columns 5-7) together with that of equivalent
Hohmann impulsive maneuvers (columns 8-10, where a specific impulse of 300
s has been assumed).
Most of the transfer time is spent in the space between CoIs. For the sake of
completeness, Table 8 reports the time of flight between the origin/destination
PLO and the CoI for the optimal transfers (column 6 with the PLO index given
in column 7) as well as the minimum (column 4) and maximum (column 5) time
of flight over all the PLOs of the family.
Overall, the tour takes 629 days (1.72 years), consumes 125 kg of propellant
28
Figure 20: Contour maps of minimum ∆V(in m/s) transfers for each PLO orbit combination
in the three inter-moon segments.
29
and requires a velocity variation of 3.07 km/s. In the three segments, the S/C
performs approximately 80, 110 and 230 revolutions around Saturn.
The advantage of this design over circle-to-circle transfers between moons
is noticeable in all respects. The Hohmann maneuvers are obviously very fast,
but the associated propellant consumption is more than four times that of the
proposed strategy, resulting in insufficient mass budget for the S/C subsystems
and science instrumentation.
Table 7: Performance comparison (velocity variation ∆V, time of flight ∆t, mass consumption
∆m) among the locally-optimal LT inter-moon transfers between CoIs, the optimal circle-to-
circle LT trajectories between moon orbits and equivalent Hohmann impulsive maneuvers.
Segment LT between CoIs LT between moons Hohmann
∆V∆t∆m∆V∆t∆m∆V∆t∆m
(m/s) (day) (kg) (m/s) (day) (kg) (m/s) (day) (kg)
Di-Te 774 170 33.8 1309 283 56.1 1304 1.5 251
Te-En 901 188 37.3 1285 256 50.7 1281 0.8 159
En-Mi 1392 270 53.5 1656 300 59.5 1650 0.6 125
Total 3067 629 125 4250 839 166 4235 2.9 535
Table 8: Transfer time between origin or destination PLO and the CoI for all the inter-moon
transfers: CR3BP (column 1), libration point (column 2), stability type (column 3, stable for
arrivals, unstable for departure segments), minimum (column 4) and maximum (column 5)
transfer time over each PLO family, transfer time for the optimal solution (column 6) along
with its PLO index (column 7).
CR3BP LiStability (∆tCoI )min (∆tCoI )max (∆tC oI )opt Optimal
(day) (day) (day) orbit index
SDi L2Stable 1.97 4.69 4.19 25
SDi L1Unstable 1.94 4.67 2.18 6
STe L2Stable 1.32 2.86 2.32 25
STe L1Unstable 1.30 2.86 1.35 2
SEn L2Stable 0.87 2.04 1.54 25
SEn L1Unstable 0.87 2.03 1.06 11
SMi L2Stable 0.57 0.90 0.83 25
30
6. Discussion
The spacecraft has an initial mass of 1500 kg and carries a Hall effect propul-
sion system which provides 36 mN of thrust and a specific impulse of 1600 s and
is characterized by a power consumption of 640 W. This level of power can be
supplied by radioisotope thermoelectric generators, hence the trajectory is not
subject to the limitations deriving from the reduced solar radiation flux far from
Earth. The interplanetary trajectory optimizes a sequence of unpowered grav-
ity assists at Venus and Earth, along with propelled and coasting arcs between
planets. The objective is to reach Saturn with a hyperbolic excess speed of 1
km/s to unpowered capture around Saturn. The characteristic launch energy is
27.04 km2/s2, the total transfer time to Saturn is 12.34 years (from January 21st
2028 to May 24th 2040) and the mass at arrival is 1014 kg. The low approach
velocity allows to carry out the orbit insertion at Saturn through a sequence of
resonant gravity assists with Titan. The maneuver is unpowered and takes 128
days. It is followed by a powered descent towards Dione, optimized to insert the
spacecraft into a planar Lyapunov orbit around the L2equilibrium point of the
Saturn-Dione circular restricted three-body problem, where the tour of the four
moons starts. The powered transfer to Dione takes 4.31 years, consuming 312
kg of propellant. This long phase can be exploited to perform scientific observa-
tions of the wide region between the orbits of Titan and Dione and make close
approaches with Dione and Rhea. The science orbits around the four target
moons are heteroclinic and homoclinic connections between planar periodic or-
bits around the collinear equilibria L1and L2of each Saturn-moon three-body
problem. The inter-moon transfers are low-thrust trajectories patching the most
favorable conditions among the stable and unstable hyperbolic invariant man-
ifolds of the same periodic orbits used for the exploration of the moons. The
science phases require negligible amounts of propellant and can be extended by
repeating the same heteroclinic and homoclinic cycles. This possibility derives
from the autonomous character of the dynamical model used. The science orbits
offer wide surface coverage (up to ±80◦latitude) and long visibility periods (up
31
to tens of hours) of the targets. The inter-moon portion of the tour lasts 1.72
years, enabling detailed observations of the E-ring and its environment.
Excluding the science orbits (whose duration can be extended arbitrarily, as
mentioned above), the entire mission takes 18.7 years, a duration comparable
to that of Cassini/Huygens (20 years) and acceptable for a project of this class.
The proposed concept is novel because it achieves the unprecedented result of
inserting the spacecraft into orbit around the four inner moons of Saturn with
a propellant mass fraction of 62%. This outcome, which would not be feasible
with chemical propulsion technologies, is made possible by the optimal design of
low-thrust arcs, the gravitational assistance of intermediate bodies (Venus and
Earth in the interplanetary transfer, Titan in the orbit insertion at Saturn) and
the dynamical properties of the invariant structures of the circular restricted
three-body problem. The design strategy is holistic, i.e., it deals with the entire
transfer taking into account the available onboard resources, the limitations of
existing power and propulsion technologies and the desired scientific return of a
mission of this importance. The low launch mass makes the proposed concept
suitable for small-to-medium-class and even multi-spacecraft missions.
Even if the trajectory is 2D, it is realistic as it enforces phasing constraints
derived from the positions of the planets in the interplanetary leg. Moreover, it
offers a reasonable degree of fidelity because the dynamical models adopted in
every phase take into account the major perturbations. It is expected that only
small adjustments will be needed to refine the proposed solution to an n-body
model.
7. Conclusions
This study outlined a strategy to explore the four Inner Large Moons of
Saturn —Dione, Tethys, Enceladus and Mimas— in close proximity using only
low-thrust and gravity assist. This is an original concept, impracticable with
conventional approaches due to the depth of Saturn’s gravity well. The ambi-
tious science goal, inserting the spacecraft into a close orbit around each moon,
32
comes at the expense of increased mission duration. To eliminate impulsive
maneuvers completely (which would need a dual propulsion system, adding
complexity and weight) a quasi-ballistic capture via a Titan gravity assist is
required. To limit the period of the subsequent orbits, the relative velocity
upon arrival at Saturn was constrained to 1 km/s. The slow approach velocity
to the planet results in a very long interplanetary transfer compared with past
missions. Given that the goal of this work is just to demonstrate the feasibility
of the concept, this limitation is acceptable.
That being said, there is ample room for improvement. Using the Saturn hy-
perbolic excess speed as an optimization parameter would establish a trade-off
between interplanetary transfer time and the duration of the multi-gravity as-
sists with Titan. This has the potential to decrease the total mission time. Also,
shortening the interplanetary phase by extending the time in orbit around to
Saturn has some scientific value by itself. Furthermore, for the sake of simplic-
ity, the descent from Titan to Dione has been computed in a sequential fashion.
Ballistic trajectories with gravity assists are followed by a propelled phase. It
is of course possible to apply thrust between the Titan flybys to shorten the
descent. Also, gravity assists with Rhea offer the possibility to accelerate the
descent and save propellant. For simplicity, this study only considered connec-
tions between planar Lyapunov orbits of the libration points of the Saturn-moon
three-body problems. They have demonstrated great scientific potential as well
as substantially reduced cost compared with circle-to-circle inter-moon trans-
fers. Nevertheless, there is also room for improvement in this area by exploring
other typologies of libration orbits, such as halo. Finally, the multi-gravity as-
sist interplanetary phase could also be refined by considering more than three
flybys en route to Saturn. This could potentially find sequences of flybys in-
volving Venus, Earth and Jupiter with smaller departure energies and shorter
transfer times.
The fact that there is substantial room for improving this results must not
be considered a limitation of the concept. In fact, it is one of its greatest
strengths. A simplified preliminary analysis has shown that the unprecedented
33
goal of inserting an spacecraft into orbit around the four moons is possible with
existing technology. The initial results being easy to improve upon only adds
credibility to the concept, and warrants future in-detail analysis.
Acknowledgements
The authors are very grateful to Dr. David Morante for his assistance on
the use of the global trajectory optimizer and acknowledge the valuable com-
ments and suggestions of the anonymous reviewer. The work of E. Fantino,
B. M. Burhani and R. Flores has been supported by Khalifa University of Sci-
ence and Technology’s internal grant CIRA-2021-65 / 8474000413. R. Flores
also acknowledges financial support from the Spanish Ministry of Economy
and Competitiveness “Severo Ochoa Programme for Centres of Excellence in
R&D” (CEX2018-000797-S). In addition, E. Fantino received partial support
from the Spanish Ministry of Science and Innovation under projects PID2020-
112576GB-C21 and PID2021-123968NB-I00. M. Sanjurjo-Rivo acknowledges
fund PID2020-112576GB-C22 of the Spanish Ministry of Science and Innova-
tion.
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