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Cycler Trajectories in Planetary Moon Systems

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Free-return cycler trajectories repeatedly shuttle a spacecraft between two bodies using little or no fuel. Here, the cycler architecture is proposed as a complementary and alternative method for designing planetary moon tours. Previously applied enumerative cycler search and optimization techniques are generalized and specifically implemented in the Jovian and Saturnian moon systems. Overall, hundreds of ideal model ballistic cycler geometries are found and several representative cases are documented and discussed. Many of the ideal model solutions are found to remain ballistic in a zero radius sphere of influence patched conic ephemeris model, and preliminary work in a high-fidelity fully integrated model demonstrates near-ballistic cycles for several example cases. In the context of recent Cassini discoveries, the Saturn-Titan-Enceladus system is investigated in the most detail and many promising solutions result. Several of the high-energy Titan-Enceladus cyclers find immediate application as Cassini extended missions options that provide frequent low-altitude Enceladus flybys.
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AAS 07-118
PLANETARY MOON CYCLER TRAJECTORIES
*
Ryan P. Russell
and Nathan J. Strange
Free-return cycler trajectories repeatedly shuttle a spacecraft between two bodies using
little or no fuel. Here, the cycler architecture is proposed as a complementary and
alternative method for designing planetary moon tours. Previously applied enumerative
cycler search and optimization techniques are generalized and specifically implemented
in the Jovian and Saturnian moon systems. In addition, the algorithms are tested for
general use to find non-Earth heliocentric cyclers. Overall, hundreds of ideal model
ballistic cycler geometries are found and several representative cases are documented
and discussed. Many of the ideal model solutions are found to remain ballistic in a zero
radius sphere of influence patched conic ephemeris model, and preliminary work in a
high-fidelity fully integrated model demonstrates near-ballistic cycles for several
example cases. In the context of recent Cassini discoveries, the Saturn-Titan-Enceladus
system is investigated in the most detail and many promising solutions result. Several
of the high energy Titan-Enceladus cyclers find immediate application as Cassini
extended missions options that provide frequent low altitude Enceladus flybys.
INTRODUCTION
Cycler orbits offer a low propellant cost means for the exploration of multiple high interest moons such as
the Galilean moons at Jupiter, Titan, and the icy moons at Saturn. The Galilean moons have long been a top
priority for space exploration. Europa, Ganymede, and Callisto each are speculated to support large subsurface
liquid water oceans, and the proximity to the surface and the size of Europa's ocean has compelling astrobiological
implications that makes Europa the top priority of destinations on NASA’s Solar System Exploration Roadmap
§
.
Io is extremely active volcanically and has a young surface, and Ganymede is the only moon in the solar system
known to have a magnetic field. At Saturn, Titan is one of the most Earth-like bodies in the solar system because
of its complex atmosphere, methane hydrological cycle, dynamic surface, and organic signatures. Enceladus has
recently become a high priority target because Cassini identified ice crystal plumes that may originate from a
liquid source extremely close to the surface.
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Traditionally, cycler orbit theory has been applied to the Sun-Earth-Mars system for application to human
missions [1-13]. Earth-Mars cycler applications generally suffer from long repeat periods, infrequent launch
opportunities, and the risky requirement to perform hyperbolic rendezvous [14]. In contrast, the brief synodic
periods of most planetary moon pairs lead to an abundance of cycler solutions with short repeat periods and
frequent initiation opportunities. Further, the mission objectives of a planetary moon cycler are achieved en route
and during flybys only and therefore do not require hyperbolic rendezvous.
In this study cycler search techniques previously applied to the Earth-Mars case are expanded to seek
similar repeating trajectories between moons that orbit a common planet. The generalized cycler search algorithm
provides timetables for the moon tour design space through an automated global search that finds and catalogues
cycler opportunities. The cycler solutions identify feasible multi-body flyby resonances useful as stand-alone
designs or preliminary guesses for sequences of conventional planetary tour designs. The cycler trajectories can
provide critical reconnaissance and communication for planned orbiters, surface landers, and aerial vehicles while
the repeat flybys enable remote sensing and in situ science of the multiple high priority targets.
The first sections overview the existing cycler search and optimization algorithms, describe the
improvements and generalizations, and discuss the relevant moon systems and dynamical models. Next, the
*
Presented at the AAS/AIAA Space Flight Mechanics Meeting in Sedona, Arizona, Jan 29-Feb 1, 2007
Jet Propulsion Laboratory, 4800 Oak Grove Drive, M/S 301-121, Pasadena, California 91109. Ryan.Russell@jpl.nasa.gov
Jet Propulsion Laboratory, 4800 Oak Grove Drive, M/S 230-205, Pasadena, California 91109. Nathan.Strange@jpl.nasa.gov
§
http://solarsystem.nasa.gov/multimedia/downloads/SSE_RoadMap_2006_Report_FC-A_med.pdf [cited 11-30-06]
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
automated ideal (circular-coplanar) model search is applied to find cyclers for five Jovian moon pairs and the
Titan-Enceladus pair in the Saturn system. Special attention is paid to Titan and Enceladus because Cassini
discoveries have recently thrust both moons into the science spotlight. The generalized algorithms are also tested
and verified in the heliocentric system and several potentially useful Venus-Mars and Venus-Mercury cyclers are
examined. Each ideal model search results in hundreds of ballistic solutions. The solutions and characteristics are
archived and a select few of the best cases are documented and discussed. Several of the solutions are optimized
over multiple periods in a zero radius sphere of influence patched conic ephemeris model and remain ballistic. Of
the most promising solutions is the Titan-Enceladus Cycler#235 that includes 9 targeted Enceladus encounters, 45
targeted Titan encounters, full orbit rotation around Saturn (for complete lighting conditions), and near Hohmann
velocities at Enceladus during a 2.4 year flight time. Several of the higher energy Titan-Enceladus cyclers have
potential applications for Cassini extended missions. Finally, to demonstrate the transition from the patched conic
ephemeris model to a high-fidelity model, multiple legs of a select few cyclers are optimized in a fully integrated
n-body plus oblateness model.
APPROACH
The general cycler trajectory problem is to find an indefinitely repeatable pattern of ballistic trajectories
that cycle a spacecraft between two celestial bodies that orbit a common body. The repeatability requirement
makes the problem substantially more difficult than typical intermoon or interplanetary spacecraft tours. In
general, exactly repeatable cyclers only exist in simplified models. Therefore, the approach for solving the general
cycler problem is reduced to two steps:
1) Perform a broad search for true cyclers in an ideal model consisting of circular and coplanar celestial
body orbits. These ideal model cyclers are exactly periodic.
2) Evaluate and optimize multiple cycles of ideal model cyclers in more realistic models. Because the
geometry of the realistic solar system model is not exactly periodic, each cycle of a realistic cycler will be
quasi-periodic. In this study, two levels of fidelity are considered for realistic models: (a) Zero radius
sphere of influence patched conic ephemeris model, and (b) Fully integrated n-body plus oblateness
model.
To reduce the search space to a manageable size, a sub-set of cyclers called free-return cyclers are sought
where one of the two orbiting celestial bodies in the ideal model is considered massless. For the case of the Sun-
Earth-Mars system, the free-return cyclers in ideal and more realistic models are computed and analyzed in detail
in [1-12]. The current study generalizes the methods outlined in [7,9] to find free-return cyclers between any two
celestial bodies that orbit a common body with near-circular, near-coplanar orbits. A brief overview of the method
follows.
First, two bodies of interest are identified that have near-circular, near-coplanar orbits around a common
body. Next, the magnitude of the spacecraft hyperbolic excess velocity (v
) with respect to the flyby body is
chosen. Assuming the ideal circular-coplanar model, all direct free-return trajectories are calculated with the
specified v
that leave the flyby body and return ballistically within a specified maximum flight time. This
procedure is described in detail in [6]. The solutions are easily visualized as a set of points and lines on the v
globe as the example in Figure 1 illustrates. The vertical arrow represents the body velocity vector and the v
globe is the locus of the tips of all possible v
vectors that emanate from the tip of the body velocity vector. The z
direction is aligned with the body velocity and the x direction is aligned with the body position with respect to the
primary leaving the y direction to be orthogonal to the body orbit plane.
In Figure 1, solutions that return to the body after exactly an integer number of body revolutions around
the primary exist as circles on the surface of the globe and are labeled “full-rev circles.” These types of free-
returns are commonly referred to as resonant transfers. Solutions that return to the body after exactly an even
integer number of half-body revolutions around the primary exist as points on the surface of the globe. They
appear as above- and below-orbit plane pairs and are labeled “half-rev x’s” and “half-rev o’s” respectively.
Lastly all free-return solutions with transfer angles that are not integer numbers of half-body revolutions exist as
single points labeled “generic dots” and are constrained to the spacecraft orbit plane.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
side view
top view
generic dots
(non-resonant returns)
half rev ‘x’s
(odd nπ returns)
body velocity
vector
3D view
half rev ‘o’s
(odd nπ returns)
full-rev circles
(even nπ returns)
Figure 1: Example v
globe identifying all three types of possible free-return solutions:
half-revolution (odd nπ transfer angle), full-revolution (even nπ), and generic (non-nπ)
Gravity-assisted flybys provide a mechanism to connect multiple free-return trajectories. The flyby has
the effect of rotating an incoming v
vector by an angle whose magnitude is a function of the flyby altitude and the
flyby body mass. In the circular coplanar model, the incoming and outgoing v
magnitudes for all free-return
trajectories are always identical. Therefore, a tour of free-return trajectories patched by gravity-assisted flybys can
be constructed as discrete steps on the v
globe where each step represents a rotation of the v
vector. The
maximum size of each step is limited by the turn angle associated with a flyby of a specified minimum altitude.
The free-return cycler problem is then reduced to a combinatorics problem of finding repeatable patterns that step
the v
vector from free-return solution to free-return solution. More specifically we want to find all 1-leg, 2-leg,
…, n-leg patterns that require valid flybys to navigate the v
vector around the v
globe on free-return solutions
with a total combined flight time that is an integer multiple of the synodic period of the two bodies of interest. If
at least one of the free-return solutions includes a trajectory that crosses the path of the second body of interest,
then the pattern can be initiated such that an encounter occurs every period. Enumerating all of the possible
combinations of half-revolution, full-revolution, and generic returns is well documented in Ref. [7]. The process
is repeated for a full range of v
values and results in a list of exactly periodic cyclers in the ideal model with a
variety of defining characteristics.
For evaluation in a more realistic model, the ideal model solutions are propagated for several repeat
cycles in the ideal model and a homotopy method is employed to parametrically walk the solution to a zero radius
patched conic model that considers ephemeris locations of the bodies and models flybys as instantaneous body-
centered velocity rotations. A final trajectory of up to ~50 legs is sought that is ballistic in the patched conic
ephemeris model. This highly constrained trajectory optimization problem is described in detail in Ref. [9].
Finally, the resulting solutions are used as initial guesses for optimization in a high-fidelity force model
including integrated flybys, n-body perturbations, and central-body oblateness. The high-fidelity optimization is
performed with Mystic, a general-use software tool under development at the Jet Propulsion Laboratory [15].
Improving strategies for transitioning from the low- to the high-fidelity ephemeris solutions is an ongoing topic of
research and warrants detailed future study. Here, it is emphasized that the high-fidelity optimization is included
only to demonstrate reasonable v costs associated with realistic planetary moon cyclers.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Figure 2 illustrates an example of a cycler trajectory. A complete cycle is shown in detail of one of the
most promising ideal model cyclers found in this study. The cycler begins at Titan on the ‘A’ label and performs a
free-return to ‘B’ after 1.88 revs. A gravity-assist flyby places the cycler on a 1.23 rev free-return to ‘C’ and is
followed by a 2:3 and consecutive 1:2 resonant returns to complete the cycle. The targeted Enceladus encounter
occurs at ‘E’ in the middle of the fourth leg or the first 1:2 resonant free return. Each gravity-assisted flyby at
Titan, including the flyby required to patch consecutive cycles, is well above the minimum altitude of 1000 km
(the lowest flyby is 1843 km), making the cycler ballistic in the ideal model.
leg 1
leg 2
leg 3
leg 4
leg 5
A
B
C,D,F,G
E
3
One Complete Cycle
Leg 1 (AB): Titan-Titan 1.88 revs
Leg 2 (BC): Titan-Titan 1.23 revs
Leg 3 (CD): Titan-Titan 2:3 resonance* (2 Titan revs, 3 sc revs)
Leg 4 (DE and EF): Titan-Titan 1:2 resonance with intermediate Enceladus encounter after 1+ revs
Leg 5 (FG): Titan-Titan 1:2 resonance* (pictured out of plane)
Titan v
= 3.18 km/s
Enc v
= 6.04 km/s
Min Titan flyby alt=1,840 km
Min Saturn distance=209,000 km
Period = 97 days
Titan orbit
Enceladus orbit
*out of plane crank angle is degree of freedom
Figure 2: Three-dimensional diagram of one cycle of the ideal model Titan-Enceladus Cycler#235
IMPROVED CYCLER SEARCH CAPABILITIES
The search and optimization techniques from [7] and [9] are generalized to deal robustly with the vastly
different time and distance scales associated with interplanetary and intermoon cyclers, including the ability for
the flyby body to be inside or outside the orbit of the target body. The following subsections highlight the most
important capabilities enabled by the current study.
Non-resonant free-return calculation
Given a specified v
and M
max
where M is the floor of the number of flyby body revolutions, an algorithm
is sought that provides all existing non-resonant return trajectories (or generic dots from Figure 1). The approach
from [6] is based on a computationally expensive table look-up method while the new approach relies on a one
dimensional root solving procedure and requires less memory storage and computational effort [16]. Figure 3
shows the period vs. flight time and v
for all direct non-resonant free returns to a body in a circular orbit with M
3 calculated from the multiple revolution Lambert problem [17,18,19]. It is emphasized that both the long and
short period direct solutions from the classical Lagrange formulation of the Lambert problem [17,18] are reported
in Figure 3. However, the inbound vs. outbound specification indicates the sign of the initial flight path angle and
is unrelated to the short vs. long period specification. The objective is to obtain an algorithm that finds all
intersections of a horizontal line of constant v
with the solution curves from the lower half of Figure 3 (for
arbitrary v
and M
max
).
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Figure 3: All direct non-resonant free returns to a body in
circular orbit for M 3
Figure 4: Zoomed view of Figure 3
Figure 5 gives an algorithm that reduces the problem to a one dimensional root-solving problem that
requires iteration on the period, T, of the transfer ellipse. Note that all equations are normalized such that the
circular radius of the flyby body is one length unit (LU) and the gravitational parameter of the primary is unity.
The algorithm is restricted to elliptical direct orbits and therefore 0 < v
< 1+ v
escape
(where 1 is the velocity
magnitude of the flyby body and v
escape
= 2
½
).
valid for direct solutions with
(
)
012v
<<+
()
2
2
min
1vv
=−
(
)
2
min min
12av=−
3
min min
2Ta
π
=
FOR
M=1, M
max
(
)
max
21TOF M
π
=+
max max min
int( ) 1NTOFT=−
FOR N=1, N
max
()
3
min
max 2 , 2 1
lb
TaMN
ππ
=+


(
)
max
max ,1
ub
TTOF N=
inout={inbound, outbound}
FOREACH inout
Root solve
(
)
,,, , 0Zv MNinoutT
=
where
lb ub
TTT<< if solution(s) exist
IF (successful) THEN record {M,N,T,inout} and other solution properties
END inout
END N
END M
Figure 5: Algorithm for calculating all non-resonant returns with a
specified v
and maximum flight time
From Figure 1 the spacecraft velocity, v, after the flyby is minimized when the v
vector is aligned
opposite to the body velocity. The second line of the algorithm in Figure 5 calculates the minimum transfer semi-
major axis, a
min
, from the conic equation and the minimum period follows on the third line. The maximum
transfer flight time, TOF
max
, is calculated noting that one normalized flyby body period is 2π time units (TU). For
a given M, the maximum floor of the number of transfer revolutions (N
max
) is calculated assuming the smallest
possible transfer period and the largest possible flight time. For a given M and N, lower and upper bounds on the
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
transfer period guess are derived from inspection for direct transfers and expressed as T
lb
and T
ub
respectively.
Next, T is iterated to seek solutions to the equation, Z=0, for both inbound and outbound transfers. If a solution
exists, then T
lb
< T < T
ub
.
The highly nonlinear equation for Z is calculated as follows. Equations (1) and (2) give the transfer semi-
major axis and departure velocity from two-body dynamics.
()
23
2aT
π
=
(1)
21va=−
(2)
Equation (3) then provides the transfer eccentricity as a function of v
. This expression is related to
Tisserand’s condition and details of the derivation are found in [16].
()
()
2
22
114evva
=− +
(3)
Equation (4) gives the absolute value of the departure true anomaly from the conic equation and Eq. (5)
expresses this as eccentric anomaly.
(
)
12
cos 1aae e
υ
=−
(4)
()()()
1
tan 2 tan / 2 1 1
E
ee
υ
=−+
(5)
The short time to periapse then is
(
)
(
)
sin 2 /
p
tEee T
π
=−

(6)
Finally, Z(M, N, v
, inout) is expressed in Eq. (7) as the difference between the flight times of the
spacecraft and the flyby body. For further details, see [16].
()
()
()
22
if inbound
if outbound
12 2 1
p
p
TN t M
inout
Z
inout
TN t M
πυπ
πυπ

+− +

=


=
=

+− +



(7)
Over the valid range of T
lb
< T < T
ub
, Z is imaginary in some regions and suffers from multiple sign
changes in the first and second derivatives with respect to T. One solution method for solving Z(T)=0 is to march
through the valid range of T and initiate a gradient based root-solver when Z is real and a sign change is detected.
Note that a few rare cases are observed where Z has two roots for fixed M, N, and inout. (See the double valued v
in Figure 4.) However, from observation, the double valued solutions are near the impractical eccentricity of 1.
An example application of the algorithm finds 220 non-resonant direct free-returns with v
= 0.5 LU/TU when
considering M 9.
For the case of Titan as the flyby body, all non-resonant retrograde transfers with v
< 1.5 LU/TU have
periapse smaller than the radius of the outermost obstructive rings at 2.92 Saturn radii. If the resonant inclined
orbits are considered, a few non-impact retrograde orbits begin to exist for v
> 1.12 LU/TU. Note that the end of
mission Cassini v
is expected to be ~1.1 LU/TU. It is therefore not restrictive to ignore retrograde solutions for a
cycler Cassini extended mission. While it is possible to construct cyclers with retrograde orbits, it is generally not
beneficial because the large v
values reduce the available flyby bending angles and thus require prohibitively long
cycle repeat periods. Despite its impracticality for the current application, to modify the algorithm in Figure 5 to
account for retrograde orbits, the adjustments in Eqs. (8) and (9) are necessary. To implement, the N=0 case
would require special attention to account for the near parabolic outbound solution with no upper bound on the
period.
max max min
int( )NTOFT=
(8)
()
22
if 0
if 0
ub
T
MN
N
N
π
=
+
=
>
(9)
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Transitioning ideal model solutions to a patched conic ephemeris model
An improved approach for the homotopy method is implemented to parametrically “walk” an ideal model
solution into a multiple cycle solution in a more realistic model. A previous approach [9] relies on the mean
orbital elements of the orbiting celestial bodies where successive sub-problems are solved by slowly increasing the
eccentricities and inclinations from circular-coplanar values to the true values, and the last sub-problem is solved
using ephemeris body locations. The technique works well in the case of planets such as Earth and Mars where
reality is closely modeled by Keplerian motion. However, the method struggles in the case of planetary moons
where perturbations from other moons and central-body oblateness cause the motion to be poorly predicted using
mean orbital elements.
The new approach parametrically walks the solution from the patched conic ideal model to a patched
conic ephemeris model using several sub-problems where the location of the orbiting bodies are determined by a
linear interpolation between the ideal model and ephemeris locations. Thus, for successive sub-problems, the
model slowly morphs from a circular coplanar model into an ephemeris model. The approach is robust and easy to
implement because it does not rely on user supplied mean elements for either body.
MODELS
The generalized free-return cycler search is applied to each of the ideal model systems in Table 1 where
the primary is considered inertially fixed and the flyby and target bodies are in circular-coplanar orbits around the
primary.
Table 1: Ideal models considered
Primary Free-return flyby body Target body
Sun Earth Mars
Sun Venus Mercury
Sun Venus Mars
Jupiter Ganymede Io
Jupiter Ganymede Europa
Jupiter Ganymede Callisto
Jupiter Europa Ganymede
Saturn Titan Enceladus
Figure 6: Flyby geometry at Enceladus
The Sun-Earth-Mars case is used to test and calibrate the improved methods noting that solutions to this
system are well documented in [7,9]. The Sun-Venus-Mercury case is investigated as a mechanism to achieve
multiple encounters with the two infrequently visited inner planets. The Sun-Venus-Mars case is evaluated in
response to a recent study suggesting Venus-Mars cyclers as an alternative architecture to transport crew and cargo
to Mars [20]. Further, the heliocentric cyclers are sought to test the general capability of the methods to work
across different time and distance scales.
All five Galilean moons of the Jovian system are of high science interest and are capable of enabling
significant gravity-assisted flybys. While the target body in the idealized model is assumed to be massless, a free-
return cycler connecting two massive bodies is possible as long as the flyby altitudes at the target body are
sufficiently high. While the higher flyby altitudes may be less desirable in terms of science, the extra degree of
freedom enabled by the inclusion of target body flybys will prove useful when transitioning to more realistic
models. The five Jovian cycler systems investigated are representative of the body pairs of highest scientific
interest. Europa and Io are perhaps the most scientifically attractive bodies, however Jupiter’s radiation
environment prohibits long spacecraft exposure in their vicinities. Ganymede is the largest body and experiences
relatively benign radiation in comparison to the closer moons. Therefore, Ganymede is chosen as the flyby body
for cyclers to each of the other moons. To improve the frequency and altitudes of the Europa encounters, a
Europa-Ganymede cycler is also considered. Note, the Ganymede-Callisto cyclers will experience the least
radiation dose of all the Jovian cyclers.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Finally, the Saturn-Titan-Enceladus system is perhaps the best planetary moon application for the free-
return cycler theory because both moons are of high science priority and the assumptions of the idealized model
are exceptionally accurate. The Titan and Enceladus orbits are nearly circular and coplanar, and Titan is capable
of providing the gravity-assists as it is the largest moon in the Saturn system by several orders of magnitude.
Furthermore, the small mass of Enceladus validates the massless assumption of the target body in the ideal model,
thus enabling arbitrarily low flyby altitudes and excellent opportunities for science. Figure 6 illustrates how the
hyperbolas at Enceladus are effectively straight lines as the mass is insufficient to significantly bend the
trajectories. Low altitude flybys are therefore possible with any B-plane angle as a degree of freedom. The
opportunities to begin a cycler re-occur frequently because the synodic period of Titan and Enceladus is a very
brief 1.5 days.
Table 2 reports the parameters for each relevant body. It is emphasized that the gravitational parameter
of the target body (for example Enceladus in the Saturn-Titan-Enceladus system) is not required for the ideal
model free return cycler search. However, the gravity of the target body is considered when transitioning ideal
model solutions to more realistic models. The patched conic ephemeris model optimization relies strictly on
ephemeris files
1
for body locations and is independent of mean orbital elements. The inclinations and
eccentricities are reported in Table 2 only to indicate the validity of the circular-coplanar assumptions of the
idealized models. With the exception of Mercury and perhaps Mars, the circular-coplanar assumptions are
reasonable for each of bodies considered. While the mean eccentricities and inclinations of the planetary moons
are generally low, it is emphasized that the n-body perturbations and the oblateness effects of the primary are
revealed in non-trivial perturbations to the osculating Keplerian orbital elements.
Table 2: Body Parameters
Body
Gravitational parameter
(km
3
/s
2
)
Radius
(km)
Ideal model circular
orbit period (s)
Ephemeris model
mean eccentricity
a
Ephemeris model mean
inclination
a
(deg)
Sun 1.3271244e11 696,000 primary Primary primary
Jupiter 126,686,535 71,492 primary Primary primary
Saturn 37,931,208 60,268 primary Primary primary
Mercury 22,321 2,440 7,600,552 0.206 7.00
Venus 324,860 6,052 19,414,153 0.0067 3.40
Mars 42,828.3 3,399 59,354,429 0.0933 1.85
Io 5,959.92 1,827 152,854 0.0041 0.036
Europa 3,202.74 1,561 306,822 0.0094 0.469
Ganymede 9,887.83 2,634 618,153 0.0011 0.170
Callisto 7,179.29 2,408 1,441,931 0.0074 0.187
Titan 8,978.14 2,575 1,377,684 0.0288 0.280
Enceladus 6.95 256.3 118,387 0.0047 0.009
a
http://ssd.jpl.nasa.gov [cited 15 Dec 2006]
All patched conic flybys referred to in this study are modeled with the assumption that they occur
instantaneously and the radius of the flyby body’s sphere of influence is zero. Note that other more realistic (and
complicated) patched conic models exist, such as those that include hyperbolic conic propagation and sphere of
influence patch point locations.
The high-fidelity force model includes the point mass gravity for the Sun, Jupiter, and Saturn. The
example Titan-Enceladus cyclers include Saturn oblateness and point mass gravity for the Saturnian moons with
gravitational parameters larger than 1 km
3
/s
2
. The example Ganymede-Europa cycler includes Jupiter oblateness
and point mass gravity for each of the Galilean moons. The ephemeris locations of the bodies are found from the
same ephemeris files that are used for the patched conic model
1
. The poles and prime meridians for Jupiter and
Saturn are based on the most recent data from the IAU Working Group on Cartographic Coordinates and
Rotational Elements of the Planets and Satellites [21].
1
URL: http://naif.jpl.nasa.gov/naif/spiceconcept.html [cited 15 Dec 2006].
URL: ftp://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/de414.bsp [cited 15 Dec 2006].
URL: ftp://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/satellites/jup230.bsp [cited Dec 15 2006]
URL: ftp://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/satellites/a_old_versions/sat242.bsp [cited 15 Dec 2006]
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
IDEALIZED MODEL CYCLER SEARCH
Each ideal model cycler search results in hundreds of variations of cyclers with a variety of defining
characteristics. While the solution sets are too large to document here, the complete trajectories are archived for
future use. A sampling of previously undocumented promising ideal model cyclers is summarized in the following
figures and tables. Figure 7 and Figure 8 illustrate the trajectories for full cycles of selected promising solutions,
where each cycle begins on the dot and ends on the cross and the numbers in the parenthesis of each title indicate
the flyby body v
in km/s and the solution ID respectively.
Figure 7: Representative heliocentric and Jovian ideal model ballistic cyclers
Table 3: Characteristics of promising heliocentric and Jovian ballistic ideal model cyclers
Body A Body B ID
Synodic
period
(day)
v
A
(km/s)
v
B
(km/s)
Number
of legs
Period
(day)
Petal
period
(yr)
Min flyby
alt. at Body A
(km)
Min dist.
to primary
(km)
Max dist.
to primary
(km)
Transit
AB
(day)
Transit
BA
(day)
Venus Mars 45 333.9 8.22 12.96 1 667.8 -65.68 19,784 108,067,501 341,571,371 554.89 112.96
Venus Mercury 22 144.6 6.62 8.61 1 433.7 -16.99 3,322 56,590,949 109,421,190 222.09 211.61
Venus Mercury 69 144.6 8.03 10.68 2 722.8 9.13 6,111 54,928,670 125,249,833 204.09 219.07
Venus Mercury 75 144.6 10.59 14.57 3 1,156.5 21.54 5,108 49,210,147 130,112,776 196.25 216.40
Europa Ganymede 93 7.05 2.37 4.10 3 28.2 -1.33 1,293 670,200 1,459,911 5.68 8.42
Europa Ganymede 131 7.05 2.40 4.10 2 21.2 -1.33 1,113 669,299 1,459,266 5.65 8.40
Europa Ganymede 159 7.05 2.45 4.11 3 28.2 -1.33 1,261 667,983 1,459,994 5.63 8.37
Ganymede Callisto 1 12.52 3.18 3.26 3 37.6 0.41 247 826,589 2,415,871 9.90 2.61
Ganymede Callisto 5 12.52 3.24 3.34 2 37.6 0.41 328 821,915 2,390,844 10.14 16.67
Ganymede Europa 5 7.05 1.66 2.57 1 35.3 -1.33 1,819 633,307 1,080,067 12.14 23.11
Ganymede Europa 43 7.05 1.87 3.89 1 14.1 -1.33 8,861 564,558 1,072,330 6.52 7.58
Ganymede Europa 316 7.05 3.20 3.81 4 49.4 -1.33 1,447 592,969 1,496,829 7.60 12.23
Ganymede Io 53 2.35 3.90 9.85 2 21.2 -1.33 518 280,283 1,075,918 8.65 5.54
Ganymede Io 185 2.35 3.97 9.90 6 49.4 -1.33 603 277,384 1,080,403 8.62 5.50
Ganymede Io 403 2.35 4.29 4.34 2 56.4 -1.33 540 412,959 1,336,522 17.14 12.63
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Figure 8: Representative Titan-Enceladus ideal model ballistic cyclers
Table 4: Characteristics of promising ballistic Titan-Enceladus ballistic ideal model cyclers
ID
v
T
(km/s)
v
E
(km/s)
Number
of legs
Period
(day)
Petal
period
(yr)
Min flyby
alt. at Titan
(km)
Min dist.
a
to Saturn
(km)
Max dist.
to Saturn
(km)
Transit
TE
(day)
Transit
ET
(day)
37 2.50 5.17 2 94.4 -3.34 1,377 219,772 1,296,179 39.85 40.04
145 2.78 7.33 3 94.4 -3.34 1,067 185,227 1,335,036 31.94 32.23
183 2.98 4.66 3 64.5 4.16 3,304 227,827 1,441,656 10.65 5.29
207 3.03 5.08 4 63.0 -3.34 3,905 192,449 1,372,495 10.60 5.34
217 3.03 5.08 4 63.0 -3.34 10,059 192,449 1,372,495 5.17 10.78
227 3.17 5.98 3 80.9 2.90 1,183 209,397 1,555,251 10.50 5.45
231 3.18 6.04 4 97.4 2.41 2,140 208,590 1,637,907 10.49 5.46
235 3.18 6.04 5 97.4 2.41 1,843 208,590 1,637,907 10.49 5.46
314 3.59 6.88 2 49.5 1.32 1,218 196,276 1,673,806 19.52 10.77
370 3.69 4.15 3 97.4 2.41 1,343 221,429 1,798,164 21.54 26.30
492 4.63 5.61 4 97.4 2.41 4,085 220,423 2,494,396 14.67 14.49
510 5.03 7.27 2 80.9 2.90 1,852 196,399 2,531,474 15.81 29.31
539 5.14 8.08 4 97.4 2.41 1,821 183,032 2,898,708 29.36 15.84
552 5.16 7.69 2 112.4 6.06 3,784 189,939 2,431,736 30.66 30.39
572 5.28 4.69 3 112.4 6.06 1,798 201,896 2,899,479 56.37 19.99
586 5.43 6.68 2 112.4 6.06 3,874 208,262 2,899,438 73.04 19.50
594 5.55 5.76 3 127.4 -38.20 6,348 210,686 2,944,806 22.31 22.14
602 5.55 5.76 4 127.4 -38.20 1,469 168,068 3,563,396 22.31 22.14
624 5.79 7.97 3 95.9 15.77 1,961 189,020 3,606,073 22.49 22.23
631 5.91 7.49 3 127.4 -38.20 4,243 197,913 3,625,914 21.66 45.42
a
For the Titan-Enceladus cyclers, Saturn’s G rings pose a hazard out to r ~176,000 km. Synodic period=1.50 days
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Table 3 and Table 4 report important metrics that characterize and define each free-return cycler solution
illustrated in Figure 7 and Figure 8. The short synodic periods of the planetary moon systems lead to a large
number of solutions and the multiple timing variations of each solution lead to a vast design trade space when
comparing solutions for different applications. The following paragraphs discuss the important cycler
characteristics and design trades associated with each of the columns of Table 3 and Table 4.
The v
values at the flyby body and the target body are important measures of energy. Depending on the
source of the cycler spacecraft, the cost of initiating a cycler can increase or decrease with v
. The initiation cost
for an Earth-Mars cycler increases with Earth v
because the cycler must be initially launched from Earth [5]. On
the other hand, a planetary moon cycler will often be more expensive to initiate for lower values of flyby body v
because the cycler spacecraft initiates from an interplanetary trajectory ending in a high speed hyperbolic
approach.
To insert into a planet-centric cycler from an interplanetary trajectory, propulsive (or perhaps
aerocapture) maneuvers are required to capture around the planet and ultimately achieve the correct moon-
centered v
. The rendezvous cost from the cycler to the flyby or target body scales directly with v
. Science
measurements during a flyby of either body will be directly impacted by the body centered velocities. Depending
on the instruments and applications, it may be desirable to perform flybys with both large and small v
values.
The ‘Number of legs’ column in Table 3 and Table 4 indicates the number of flyby body free-return
trajectories, where one of the legs includes an encounter with the target body. Thus, a cycler with n legs has n+1
flybys per cycle. For science purposes, it is desirable for a cycler to include as many flybys as possible.
The cycler period is the combined time of flight of each leg and is constrained to be an integer multiple of
the system synodic period. Because there is only one guaranteed encounter with the target body per full cycle, the
cycler period dictates the frequency of target body encounters. As [7] illustrates, the number of cycler solutions
increases dramatically as the allowable period increases. Further, the enumerating technique becomes practically
infeasible when the allowable cycler periods are too long. For reasonable computation run-times (less than a few
days on one modern processor), a general rule of thumb is to seek solutions with periods less than or equal to 8
flyby body periods. This rule limits the number of potential cycler legs to approximately 10. Less exhaustive
searches such as the one described in [5] can substantially increase the number of potential legs and the cycler
period, however these solutions do not include nπ transfers when n > 2. To emphasize the importance of short
delays between successive target body encounters, the enumerating technique based on [7] is used exclusively for
the present study.
The petal periods reported in Table 3 and Table 4 are a measure of the angular difference in locations of
the flyby body between consecutive cycles. If the trajectories are considered ‘petals’ on a flower that is centered
at the primary, the petal period is the time required to complete the flower with 2π radians of petals. This metric
is most important for the planet-centric cyclers with flyby bodies that are synchronously locked with their orbital
rates. For the synchronously rotating body, the sequence of v
-
and v
+
vectors associated with each flyby
remains unchanged in the body-fixed frame for successive periods. (The only exception is the case of a resonant
flyby where there is a potential degree of freedom in choosing the placement of v
along the full-rev circle as
illustrated in Figure 1.) Therefore, in the idealized model, the ground tracks of each flyby of the gravity-assist
body will be identical for successive cycles. For example, if there are four flybys of the gravity-assist body during
each cycle, then the set of the four ground tracks will be repeated every cycle (again with the exception of any
resonant flybys with a degree of freedom). Of course, when perturbations of a realistic model are considered, the
ground-tracks are expected to be quasi-periodic rather than exactly repeating as in the ideal model.
Because the periods of the planet-centric cyclers are short compared to the orbital period of the primary,
the sun-line direction is approximately fixed. Consequently, the lighting conditions for science measurements
during each flyby will change as the orientation of the petals rotate around the primary. The petal period is
therefore the approximate time required to achieve all possible lighting conditions for each of the n+1 flybys on a
given cycler. For the planet-centric cyclers with synchronously rotating bodies, a short petal period is highly
desirable. Note, the sign of the petal period indicates the direction of rotation.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
The minimum and maximum distances to the primary may be useful for filtering cyclers that exceed a
desired range of operation. The Saturn centric cyclers require a conservative minimum distance of 2.92 Saturn
radii (~176,000 km) to avoid harmful ring particle collisions.
The final two columns of Table 3 and Table 4 report the transit times involving the target body. As
mentioned previously, each ideal model cycler consisting of free-return trajectories to the flyby body can be
initiated at multiple times such that an encounter with the target body is ensured. The transit leg trajectory will
intersect the path of the target body once on an inbound path and once on an outbound path (except in the limiting
case where the periapse of the transit is the radius of the target body orbit). Further, the inbound and outbound
encounter opportunities are repeated for each revolution of a multiple revolution transit leg. It is emphasized that
multiple pairs of transit times are possible for each cycler reported in Table 3 and Table 4, although only one pair
is listed. For operational purposes it is difficult to schedule consecutive flybys within 3 or 4 days (and transit
times greater than 10 days are preferable). For the fast moving systems such as the Titan-Enceladus system, a
minimum transit time of 3 days is an important constraint because many of the favorable cycler solutions fail to
have any transit time pairs that meet this requirement. Note that transit time pairs with near equal magnitudes will
generally lead to improved convergence properties when transitioning the solution to an ephemeris model.
Table 5 and Table 6 provide the formal nomenclature associated with each or the reported ideal model
cyclers. The purpose of the naming system is to provide an efficient means of describing these complicated
trajectories uniquely. Given the definitions outlined in [8] and a descriptor string for each leg of an arbitrary
cycler, all of its characteristics can be calculated, and the entire trajectory can be systematically reproduced. Note
that each leg begins with an h, f, or g representing a half-revolution (odd nπ), full-revolution (even nπ), or a
generic (non-nπ) free-return respectively. A capital letter indicates the leg that includes the target body encounter.
Table 5: Formal nomenclature for cyclers from Table 3
ID Leg 1 Leg 2 Leg 3 Leg 4 Leg 5 Leg 6
VenMar#45 G(2.97216,349.97729,U) - - - - -
VenMer#22 G(1.93012,1054.84284,U) - - - - -
VenMer#69 g(1.33364,480.11127,Ls) G(1.88322,1037.96013,U) - - - -
VenMer#75 g(1.31055,471.79942,U) G(1.83643,1021.11483,U) f(2:3,74.20508,0.00960) - - -
EurGan#93 g(1.97030,349.30891,U) G(3.97176,709.83427,U) f(2:1,88.69348,0.57296) - - -
EurGan#131 G(3.95655,704.35739,U) f(2:1,87.95239,90.00000) - - - -
EurGan#159 G(3.94206,699.14318,U) f(2:1,87.24509,120.26932) f(2:1,87.24509,59.73068) - - -
GanCal#1 G(1.74871,269.53421,U) g(1.50246,540.88534,L) f(2:1,77.40130,0.03291) - - -
GanCal#5 g(1.50425,541.53130,L) G(3.74691,628.88825,U) - - - -
GanEur#5 G(4.92758,2493.92898,U) - - - - -
GanEur#43 G(1.97103,1069.57159,U) - - - - -
GanEur#316 g(1.31322,472.76044,U) h(1.5,540.0,L,-3.98557) g(1.31322,472.76044,U) G(2.77217,1357.97970,U) - -
GanIo#53 g(0.97232,710.03419,U) G(1.98423,1434.32320,U) - - - -
GanIo#185 g(0.96288,706.63724,U) f(1:2,84.97911,89.99999) g(0.96288,706.63724,U) f(1:2,84.97911,57.73932) G(1.97285,1430.22609,U) f(1:2,84.97911,90.00890)
GanIo#403 g(3.72334,1700.40403,Ll) G(4.16078,2217.88234,L) - - - -
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Table 6: Formal nomenclature for cyclers from Table 4
ID Leg 1 Leg 2 Leg 3 Leg 4 Leg 5
TitEnc#37 g(0.91247,688.48877,U) G(5.01017,3963.66276,L) - - -
TitEnc#145 g(0.89794,683.25735,U) G(4.02471,3248.89419,L) f(1:2,64.90258,142.00153) - -
TitEnc#183 g(0.89055,680.59930,U) g(2.15189,1134.67873,U) F(1:2,61.35060,180.0) - -
TitEnc#207 g(0.88869,679.92726,U) g(1.05974,741.50710,Ll) F(1:2,60.31038,180.0) f(1:2,60.31038,167.71581) -
TitEnc#217 g(0.88869,679.92726,U) F(1:2,60.31038,0.0) g(1.05974,741.50710,Ll) f(1:2,60.31038,-179.90781) -
TitEnc#227 g(0.88489,678.56001,U) g(3.19166,1508.99845,U) F(1:2,57.91040,180.0) - -
TitEnc#231 g(0.88468,678.48383,U) g(1.22599,441.35506,U) f(3:5,55.34820,167.71581) F(1:2,57.76202,180.0) -
TitEnc#235 g(0.88468,678.48383,U) g(1.22599,441.35506,U) f(2:3,55.18988,179.99996) F(1:2,57.76202,180.0) f(1:2,57.76202,154.65857)
TitEnc#314 g(1.20284,433.02184,U) G(1.89950,1403.81944,Ls) - - -
TitEnc#370 g(0.87690,675.68567,Ls) g(2.23376,804.15321,U) F(3:5,48.53243,180.0) - -
TitEnc#492 G(1.82913,1018.48816,L) f(1:1,40.91079,0.00006) g(2.28153,461.35072,U) f(1:1,40.91079,179.99987) -
TitEnc#510 g(2.24732,449.03519,U) G(2.82923,1378.52326,L) - - -
TitEnc#539 g(1.27588,99.31575,U) f(1:1,35.08563,179.99992) G(2.83479,1380.52313,L) f(1:1,35.08563,-0.00001) -
TitEnc#552 g(3.22220,799.99089,U) G(3.82857,1738.28475,L) - - -
TitEnc#572 g(1.26162,94.18203,U) G(4.78915,1724.09360,L) f(1:1,33.41047,-0.00438) - -
TitEnc#586 g(1.24728,89.02223,U) G(5.80348,2089.25340,L) - - -
TitEnc#594 g(2.20335,433.20509,U) G(2.78752,1003.50729,L) f(3:2,34.01307,0.00040) - -
TitEnc#602 g(2.20335,433.20509,U) f(1:1,30.24411,-176.67648) G(2.78752,1003.50729,L) f(2:1,36.04284,0.00001) -
TitEnc#624 g(1.21211,76.36107,U) G(2.80454,1009.63414,L) f(2:1,33.39447,-0.01457) - -
TitEnc#631 g(1.78380,642.16785,L) G(4.20707,794.54453,U) f(2:1,32.10391,179.99994) - -
PATCHED CONIC EPHEMERIS MODEL OPTIMIZATION
Multiple cycles of a few example cyclers are transitioned to the patched conic ephemeris model and
illustrated in Figure 9 - Figure 13. Remarkably, many of the cyclers remain ballistic. The Venus-Mars #45
solution in Figure 9(a) easily converges to ballistic and is similar to the Earth-Mars Aldrin cycler [3,13] because it
consists of just one multiple revolution non-resonant return. Despite several promising Venus-Mercury ideal
model cyclers such as #22 and #45 presented in Figure 7, no ephemeris solutions are presented because the large
eccentricity of Mercury proved to high for multiple cycles to remain ballistic in the ephemeris model. The
Ganymede-Callisto #1 solution from Figure 9(b) is a representative ballistic solution that enjoys very low radiation
exposures and an extremely short petal period (~0.41 years from Table 3) as indicated by the completion of ~2.5
rotations around Jupiter over the course of 10 cycles. The Ganymede-Europa #316 ballistic solution from Figure
10(a) is noteworthy because of its many legs, the large out-of-plane motion, and the inclusion of a 3π transfer
(2
nd
leg from Table 5). The Europa-Ganymede #131 solution from Figure 10(b) is a relatively simple cycler that
enables 2 Europa flybys and 1 Ganymede flyby every 21 days, and the radiation exposure is minimized because it
never goes inside of Europa’s orbit.
Figure 11 - Figure 13 give examples of Titan-Enceladus patched conic ephemeris cyclers. Of the low v
solutions, #235 from Figure 12(b) is particularly promising. This example includes 45 Titan flybys, 9 Enceladus
flybys, and a flight time of 2.4 years. The Enceladus encounters are equally spaced around Saturn enabling the
full possibilities of lighting conditions, and the trajectory easily converges to ballistic due to the large flyby
altitudes in the ideal model solution. Further, as is common for all planetary cyclers, initiation opportunities are
abundant because of the short synodic period (1.5 days for Titan-Enceladus system).
Note that all the free-return cyclers are guaranteed to have exactly one target-body encounter per cycle.
However, the fast Enceladus period leads to the possibility for many untargeted flybys, especially when the transit
leg includes multiple revolutions. Close inspection of the Cycler#235 in Figure 12(b) shows 24 additional
encounters with close approaches less than 200,000 km while 4 of those are less than 100,000 km. As a second
The odd-nπ solutions are sought in all of the ideal model cycler searches in this study. However, unlike the Earth-Mars case [7], ballistic
solutions that include odd-nπ free returns are much less common for the planet-centered cyclers considered.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
example, the transit leg on the #37 cycler from Figure 11(a) is an 11+ revolution transfer. Thus, in addition to the
8 targeted Enceladus flybys, there are 17 untargeted Enceladus flybys with close approaches less than 100,000 km
where 9 of those are less than 25,000 km. The frequency and proximity of the untargeted flybys are highly
sensitive to a specific epoch.
Table 7 gives the pertinent data for the example patched conic ephemeris model Cycler#235 shown in
Figure 12(b). While the Titan-Enceladus system is circular coplanar to first order, a quick comparison between the
times and v
values of Table 7 and Table 4 illustrates the non-trivial effects of using an ephemeris vs. a circular
coplanar model.
The v
at Titan for the Cassini spacecraft has spanned the approximate range of 5.8 6.0 km/s.
Considering that small leveraging maneuvers near apoapse [16] can significantly reduce flyby body v
, the ideal
model Titan-Enceladus cyclers with v
> ~5 km/s (see Table 4) are reasonable for extended mission consideration.
Of the high energy cyclers, those in Figure 13 are illustrated because of their short petal periods leading to
favorable rotation rates around Saturn. The out of plane component of solution #572 from Figure 13(a) is also
attractive; however, as the title indicates, this 562 day trajectory requires 222 m/s v and fails to remain outside of
Saturn’s G ring radius of 176,000 km. The #586 trajectory from Figure 13(b) remains outside the G ring and
requires 154 m/s to achieve 5 Enceladus and 10 Titan encounters in 561 days. Because the higher v
trajectories
are dynamically more constrained, the patched conic ephemeris optimization outlined in [9] converges to ballistic
solutions much easier for trajectories with low v
. Noting that the method in [9] has only a limited ability to
minimize v for non-ballistic trajectories, it is anticipated that detailed optimization of any one trajectory with a
high-fidelity optimizer will reduce v requirements.
(a)
(b)
Figure 9: Example Venus-Mars and Ganymede-Callisto ephemeris patched conic cyclers
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
(a)
(b)
Figure 10: Example Ganymede-Europa and Europa-Ganymede ephemeris patched conic cyclers
(a)
(b)
Figure 11: Example low v
Titan-Enceladus ephemeris patched conic cyclers, Part I
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
(a)
(b)
Figure 12: Example low v
Titan-Enceladus ephemeris patched conic cyclers, Part II
(a)
(b)
Figure 13: Example high v
Titan-Enceladus ephemeris patched conic cyclers (applicable to Cassini extended missions)
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Table 7: Details
a
on the patched conic ephemeris Titan-Enceladus Cycler#235 from Figure 12(b)
Leg
#
Start
(T or E)
TOF
(day)
v
(km/s)
RA
(deg)
DEC
(deg)
Leg
#
Start
(T or E)
TOF
(day)
v
(km/s)
RA
(deg)
DEC
(deg)
1 T 14.187 3.103 105.17 -25.25 28 T 10.507 3.246 -43.35 15.75
2 T 19.570 3.145 170.61 0.81 29 E 5.444 5.951 -6.82 -3.17
3 T 31.891 3.115 137.84 -20.69 30 T 15.946 3.257 -32.21 29.65
4 T 10.459 3.114 153.69 -8.13 31 T 14.063 3.257 -51.66 18.92
5 E 5.489 5.506 -174.11 10.05 32 T 19.509 3.206 9.19 -10.24
6 T 15.945 3.120 155.95 -12.06 33 T 31.891 3.239 -23.19 11.67
7 T 14.118 3.120 146.02 -11.62 34 T 10.539 3.239 -8.05 -1.28
8 T 19.600 3.172 -152.35 18.14 35 E 5.403 6.901 26.67 -18.86
9 T 31.891 3.145 178.52 -6.37 36 T 15.945 3.229 3.55 12.02
10 T 10.458 3.145 -169.54 10.17 37 T 14.130 3.229 -15.92 2.62
11 E 5.494 4.499 -135.54 25.54 38 T 19.484 3.178 47.86 -24.19
12 T 15.945 3.162 -166.67 4.98 39 T 31.891 3.205 11.06 -5.46
13 T 14.051 3.162 -177.16 7.13 40 T 10.554 3.205 27.66 -17.78
14 T 19.607 3.200 -109.57 27.44 41 E 5.382 7.423 64.74 -27.85
15 T 31.891 3.187 -144.72 10.96 42 T 15.945 3.186 37.34 -3.58
16 T 10.481 3.187 -129.14 24.24 43 T 14.189 3.186 18.96 -14.66
17 E
5.473
4.185
b
-90.73 28.61 44 T 19.483 3.158 91.48 -27.16
18 T 15.946 3.211 -129.89 43.74 45 T 31.891 3.164 50.27 -13.39
19 T 14.014 3.211 -137.83 22.79 46 T 10.540 3.164 68.81 -26.96
20 T 19.590 3.216 -65.03 23.05 47 E 5.394 7.380 107.11 -25.37
21 T 31.891 3.223 -104.96 24.03 48 T 15.945 3.142 71.05 -17.91
22 T 10.485 3.223 -84.26 26.48 49 T 14.212 3.142 58.94 -26.25
23 E 5.471 4.851 -43.15 15.31 50 T 19.510 3.150 132.82 -17.26
24 T 15.946 3.250 -77.08 42.78 51 T 31.891 3.133 90.55 -15.82
25 T 14.018 3.250 -93.47 27.51 52 T 10.506 3.133 112.37 -23.47
26 T 19.550 3.221 -26.26 7.91 53 E 5.428 6.740 147.10 -10.89
27 T 31.891 3.246 -56.82 33.89 54 T 15.945 3.110 111.26 -13.18
a
The propagation is a zero radius sphere of influence patched conic and the ephemeris positions of the moons are relative to a fixed Saturn. Right ascension (RA)
and declination (DEC) are expressed in the ecliptic J2000 frame. The start date is 8774.549 days after J2000 (Jan-10-2024).
b
For orbiting or landing at Enceladus, a mission designer should target the lowest v
of all the Enceladus encounters. Note the minimum possible in the ideal
model is from the Hohmann transfer: Enceladus v
= 3.71 km/s, Titan v
= 2.39 km/s
HIGH-FIDELITY OPTIMIZATION
The assumptions of the zero radius sphere of influence patched conic ephemeris model include several
non-trivial error sources that manifest when transitioning solutions to a high-fidelity model. These error sources
include the non-zero radius of the sphere of influence, central-body oblateness, and n-body perturbations. For the
examples considered in this study, the largest of these error sources is the assumption that that the sphere of
influence for the flyby body is negligible compared to the size of the flyby orbit around the primary. A common
definition for the radius of the sphere of influence around a small body in a circular orbit around a primary is given
in Eq. (10) where d is the separation distance between the bodies and µ is the mass ratio of the smaller body to the
primary [22].
25
SOI
rd
µ
=
(10)
The patched conic assumption is clearly better in the case of the heliocentric cyclers noting that r
SOI
~0.006d for the Sun-Venus and the Sun-Earth systems compared to ~0.035d for the Saturn-Titan system, ~0.023d
for the Jupiter-Ganymede system, and ~0.014d for the Jupiter-Europa system. Further, the oblateness effects of
the primary and n-body perturbations (both of which are ignored in our patched conic ephemeris model) play a
significantly greater role in the planet-centric cyclers. For these reasons the transition to a high-fidelity force
model is substantially more difficult for the planet-centric cyclers. To demonstrate the transition, Table 8
summarizes the results of optimized single cycles of four example cyclers in high-fidelity models using the
patched conic ephemeris solutions as initial guesses.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
Table 8: Single cycles optimized in high-fidelity models
1,2,3
(a) Titan-Enceladus #183
(b) Titan-Enceladus #235 (c) Titan-Enceladus #586 (d) Ganymede-Europa #316
Encounter t (day)
flyby alt.
(km) or v
Encounter t (day)
flyby alt.
(km) or v
Encounter t (day)
flyby alt.
(km) or v
Encounter t (day)
flyby alt.
(km) or v
Titan
0.00 10,510
Titan
0.00 1,858
Titan
0.00 4,417
Ganymede
0.00 1,224
Dione 7.13 2,879 v 8.95
1.1 m/s
Tethys 1.51 11,656 v 8.78
78.7 m/s
Dione 7.36 79,911 Rhea 9.27 49,765 Enceladus 1.58 82,145
Ganymede
10.80 2,094
Mimas 7.45 166,523 v 13.55
13.2 m/s
Dione 19.15 70,850
Ganymede
20.17 4,778
Rhea 16.63 47,197
Titan
19.58 2,040 Enceladus 19.33 154,979 v 23.31
42.4 m/s
Mimas 27.26 176,755 Dione 22.08 115,461 Rhea 19.99 578
Europa
27.72 495
v 27.29
8.8 m/s
Enceladus 22.30 145,151 Enceladus 37.13 180,599
Ganymede
39.99 8,230
Enceladus 27.36 169,665 Tethys 22.32 91,913 Dione 54.75 58,404
Ganymede
49.35 1,492
Rhea 27.72 7,541 v 23.30
7.5 m/s
Enceladus 54.94 184,774
Titan
34.36 3,670 Rhea 33.38 70,927 v 54.98
6.5 m/s
v 37.10
1.4 m/s
Titan
51.49 1,000 Tethys 55.27 106,293
Enceladus 37.18 151,288 Enceladus 54.19 172,350
Enceladus
72.92 912
Mimas 44.96 129,365
Enceladus
61.92 3 v 73.08
1.6 m/s
Enceladus
45.02 3 Mimas 62.00 145,679 Dione 90.97 182,699
Mimas 45.12 87,092 Tethys 62.07 76,144 v 91.00
3.2 m/s
Titan
50.27 9,234 Mimas 62.24 155,684 Enceladus 91.16 4
Rhea 52.45 113,199 v 63.55
20.0 m/s
Tethys 91.22 191,625
Tethys 53.21 48,754
Titan
67.44 1,801
Titan
92.81 4,889
Dione 61.52 148,847 Enceladus 70.13 114,438
Titan
112.30 5,693
v 61.90
22.2 m/s
Enceladus 70.39 165,493
Titan
64.45 10,201 v 70.45
21.1 m/s
Hyperion 74.79 81,157
Mimas 78.17 62,970
Titan
83.39 999
v 94.70
0.2 m/s
Enceladus 94.78 93,155
Enceladus 95.03 127,658
Titan
97.47 2,232
1
Begin date for (a) - (d) is February 1, 2024 3:23:12; January 24, 2024 3:59:16; January 29, 2024 15:24:38; and May 3, 2019 12:17:26 respectively
2
Total v for (a) - (d) is 32, 63, 11, and 121 m/s respectively
3
The bold encounters result from the ideal model cycler geometry while the others are serendipitous and initially untargeted.
Close approaches with flyby radii less than 200,000 km to all moons with gravitational parameters larger
than 1 km
3
/s
2
are reported in Table 8. Note that only one low altitude targeted flyby of Enceladus or Europa
results from the initial guess. However, several high altitude serendipitous encounters with Enceladus occur in
each of the cases (a) –(c). In particular, note that case (c) includes an extra very low altitude Enceladus encounter
with a flyby altitude of 4 km. The case (a) Titan-Enceladus Cycler#183 includes one very low altitude (3 km)
and two high altitude Enceladus flybys; four medium altitude Titan encounters; and many untargeted moon
encounters costing a total v of 32 m/s. The case (b) Titan-Enceladus Cycler#235 includes six low altitude Titan,
one low altitude Enceladus, 6 high altitude Enceladus, and many untargeted moon encounters costing a total v of
63 m/s. Note that a second independently optimized cycle of Titan-Enceladus Cycler#235 required a total
v of
40 m/s. The case (c) Titan-Enceladus Cycler#586 includes three medium altitude Titan, two low altitude
Enceladus, 4 high altitude Enceladus, and many untargeted moon encounters costing a total v of 11 m/s. Note
the v
at Titan is ~5.75 km/s making this example applicable for a Cassini extended mission. The case (d)
Ganymede-Europa Cycler#316 includes five targeted Ganymede, one targeted Europa, and no untargeted moon
encounters costing a total v of 121 m/s. While not documented in Table 8, a high-fidelity simulation of one cycle
of the Europa-Ganymede Cycler#131 costs a total v of 58 m/s and includes an extra close flyby of Europa.
As anticipated, the patched conic ephemeris solutions with the very high flyby radii are generally more
difficult to converge to the low v solutions in the high-fidelity models because of the large violation of the zero
radius sphere of influence assumption. Future work includes seeking methods to mitigate this effect such as
altering the constants in the lower-fidelity models in order to mimic for the more realistic timing and geometry in
the high-fidelity model. In addition, improved ideal model (circular-coplanar) cyclers could be sought by
removing the massless assumption of the target body. The Jovian system cyclers in particular would benefit from
such a change.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
The design of high-fidelity moon tours is generally a time consuming process involving an artistic
combination of methods, software, and intuition. The examples included here are meant to demonstrate the
feasibility of realistic planetary moon cyclers. Detailed refinement of each reported sequence would likely lead to
improved v costs. Further, it is emphasized that the boundary conditions on each sequence are heavily
constrained so that the timing and geometry of the cycler is maintained. Relaxing these constraints is another
source that can reduce v costs. As an example, by adding one extra Titan leg and re-optimizing the sequence
presented in Table 8(a), the total v reduces from 32 to 17 m/s. This dramatic improvement is not indicative of all
cases. However, large maneuvers near the end of a sequence can often be significantly reduced by including
additional flybys. In general, patching together and optimizing multiple cycles such as the 54 leg cycler illustrated
in Table 7 and Figure 12(b) is beyond the scope of the software and intent of the present study. Future work
includes automating methods to enable such multi-cycle optimization in the high-fidelity models.
CONCLUSIONS
Existing cycler search algorithms previously applied to the Earth-Mars case are generalized and improved
for application to the planetary moon cycler problem. The short synodic periods of the planetary moon systems
significantly widens the design space for finding useful cycler trajectories. As a result, the existence of hundreds
of ideal model ballistic cyclers is demonstrated for the Galilean moon pairs at Jupiter and the Titan-Enceladus
moon pair at Saturn with repeat times ranging from ~2 to ~18 weeks. The complete database is archived and can
be queried or sorted for quick assessment of the cycler architecture and preliminary guesses for future planetary
moon mission and tour design applications.
For evaluation in a more realistic model, an improved homotopy method is implemented that seeks
multiple cycles of ballistic solutions in a patched conic ephemeris model. Notably many of the multi-cycle
trajectories that include up to 54 flybys remain ballistic and several resulting examples are documented. As a
feasibility demonstration, we optimize single cycles of four representative cyclers in a high-fidelity force model
based on initial guesses from the patched conic ephemeris model. The preliminary analysis suggests that the high-
fidelity force models often differ significantly from the zero radius sphere of influence patched conic ephemeris
model for the planetary moon systems considered. However, for the Titan-Enceladus high-fidelity examples, we
find that the v costs per flyby are similar in magnitude to Cassini, noting that the solutions are heavily epoch
dependent and further refinement would likely improve the results. Future work is required to further assess the
viability and strategies of designing realistic cyclers.
Special attention is paid to the Titan-Enceladus system because of the recent heightened science interest
due to Cassini and the exceptional accuracy of the massless Enceladus assumption. Of the low-energy Titan-
Enceladus cyclers, #235 is recommended as one of the most promising because of its short period, frequent
encounters with both bodies, multiple degrees of freedom and flyby geometries at Titan, potential for full lighting
conditions, and low approach velocities at Enceladus. The high energy Titan-Enceladus cyclers are candidates for
the Cassini extended missions that undoubtedly will require frequent, low-cost encounters with Enceladus.
The generalized free-return cycler theory provides alternative and complementary methods to explore
some of the highest priority celestial bodies according to the planetary science community including Titan,
Enceladus, and the Galilean moons at Jupiter. The repeat flybys of a cycler enable remote sensing surface science
as well as in situ measurements of atmospheres, electromagnetic fields, and plumes. The planetary cycler
trajectories can act as stand-alone flyby missions or as roadmaps of the trade space for the traditional planetary
tour design problem. Furthermore, a cycler can provide invaluable reconnaissance and act as a
telecommunications relay for surface landers, orbiters, or aerial vehicles. For a very low propellant cost and only a
modest percentage increase in total mission duration, the cycler architecture is an attractive option for maximizing
science for a variety of planetary moon missions.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
ACKNOWLEDGEMENTS
The authors thank Kim Reh, Tom Spilker, Jim Cutts, and Brent Buffington for their interest and support.
Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a
contract with the National Aeronautics and Space Administration.
REFERENCES
1
Rall, C. S., “Freefall Periodic Orbits Connecting Earth and Mars,” Ph.D. Thesis, Department of Aeronautics and
Astronautics, Massachusetts Institute of Technology, Oct. 1969.
2
Hollister, W. M., “Periodic Orbits for Interplanetary Flight,” Journal of Spacecraft and Rockets, Vol. 6, No. 4, 1969, pp. 366-
369.
3
Byrnes, Dennis V., Longuski, James M. Aldrin, Buzz, "Cycler Orbit Between Earth and Mars," Journal of Spacecraft and
Rockets, Vol. 30, No. 3, May-June 1993, pp. 334-336.
4
Nock, T., Duke, M., King, R., Jacobs, M., Johnson, L., McRonald, A., Penzo, P., Rauwolf, J., Wyszowski, C., “An
Interplanetary Rapid Transit System Between Earth and Mars,” Expanding the Frontiers of Space; Space Technology and
Applications International Forum – STAIF 2003, edited by El-Genk, M. S., Melville, NY, American Institute of Physics,
2003, pp. 1074-1086.
5
Russell, R. P., Ocampo, C. A., “Systematic Method for Constructing Earth-Mars Cyclers Using Free-Return Trajectories,”
Journal of Guidance, Control, and Dynamics, Vol. 27, No. 3, 2004, pp. 321-335.
6
Russell, R. P., Ocampo, C. A.,Geometric Analysis of Free-Return Trajectories Following a Gravity-Assisted Flyby”,
Journal of Spacecraft and Rockets, Vol. 42, No. 1, 2005, pp. 694-698.
7
Russell, R. P., Ocampo, C. A., “Global Search for Idealized Free-Return Earth-Mars Cyclers,” Journal of Guidance, Control,
and Dynamics, Vol. 28, No. 2, 2005, pp. 194-208.
8
McConaghy, T. T., Russell, R. P., Longuski, J. M., “Towards a Standard Nomenclature for Earth-Mars Cycler Trajectories,”
Journal of Spacecraft and Rockets, Vol. 42, No. 4, 2005, pp. 694-698.
9
Russell, R. P., Ocampo, C. A., “Optimization of a Broad Class of Ephemeris Model Earth–Mars Cyclers,” Journal of
Guidance, Control, and Dynamics, Vol. 29, No. 2, 2006, pp. 354-367.
10
McConaghy, T. T., Yam, C. H., Landau, D. F., Longuski, J. M., “Two-Synodic-Period Earth-Mars Cyclers with Intermediate
Earth Encounter,” AAS Paper 03-509, Aug. 2003.
11
Byrnes, D. V., McConaghy, T. T., Longuski, J. M., “Analysis of Various Two Synodic Period Earth-Mars Cycler
Trajectories,” AIAA Paper 2002-4423, Aug. 2002.
12
Niehoff, J., “Pathways to Mars: New Trajectory Opportunities,” American Astronautical Society, AAS Paper 86-172, July
1986.
13
Chen, K., McConaghy, T., Okutsu, M., Longuski, J. “A Low-Thrust Version of the Aldrin Cycler,” AIAA Paper 2002-4421,
Aug. 2002.
14
Landau, D. F., Longuski, J. M., “Guidance Strategy for Hyperbolic Rendezvous,” Paper AIAA-2006-6299, Aug. 2006.
15
Whiffen, G. J., Sims, J. A., “Application of the SDC optimal control algorithm to low-thrust escape and capture trajectory
optimization”, Paper AAS 02-208, AAS/AIAA Space Flight Mechanics Meeting, San Antonio, Texas, 2002.
16
Strange, N.J., Sims, J.A., “Methods for the Design of V-Infinity Leveraging Maneuvers,” Paper AAS 01-437, Aug. 2001.
17
Prussing, J. E., Conway, B. A., Orbital Mechanics, Oxford University Press, New York, 1993. pp.63-80.
18
Prussing, J. E., “A Class of Optimal Two-Impulse Rendezvous Using Multiple-Revolution Lambert Solutions,” Journal of
the Astronautical Sciences, Vol. 48, Nos. 2 and 3, April–September 2000, pp. 131–148.
19
Shen, H., and Tsiotras, P., “Using Battin’s Method to Obtain Multiple-Revolution Lambert’s Solutions,” AAS Paper 03-568,
Aug. 2003.
20
Turner, A., “Low Road to Mars: The Venus-Mars Cycler,” Paper AAS 07-175, Jan. 2007.
21
Seidelmann, P.K., Abalakin, V.K., Bursa, M., Davies, M.E., Bergh, C. de, Lieske, J.H., Oberst, J., Simon, J.L., Standish,
E.M., Stooke, P., Thomas, P.C., “Report of the IAU/IAG Working Group on Cartographic Coordinates and Rotational
Elements of the Planets and Satellites: 2000,” Celestial Mechanics and Dynamical Astronomy, Vol. 82, Issue 1, 2002, pp.
83–111.
22
Wiesel, W. E., Spaceflight Dynamics, McGraw-Hill, Boston, 1997. pg. 300.
cite the peer-reviewed update: Russell, R. P., Strange N.J., “Planetary Moon Cycler Trajectories,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 1, 2009, pp. 143-157.
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