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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 (A→B): Titan-Titan 1.88 revs

Leg 2 (B→C): Titan-Titan 1.23 revs

Leg 3 (C→D): Titan-Titan 2:3 resonance* (2 Titan revs, 3 sc revs)

Leg 4 (D→E and E→F): Titan-Titan 1:2 resonance with intermediate Enceladus encounter after 1+ revs

Leg 5 (F→G): 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

).

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

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)

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.

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]

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

A→B

(day)

Transit

B→A

(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

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

T→E

(day)

Transit

E→T

(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

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.

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) - - - -

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.

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

(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

(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)

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.

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.

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.

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.

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