1. Introduction
The multiple swing-by trajectory in Jupiter system is
investigated. It is denoted as gJupiter Tourh and was
first performed by NASA in Galileo mission.
To establish the orbit around Jupiter with a practical
orbital period, the Jupiter moons multiple flyby is
absolutely required, to keep the required
deterministic delta-V (i.e. amount of fuel) to the
practical amount.
The analyses in this paper have been done for the
preliminary design of the Jupiter Magnetospheric
Orbiter (JMO), which is a part of what is proposed
as the JAXA/ESA joint mission toward Jupiter in the
Cosmic Vision framework.
This paper focuses on flyby with two large Galilean
satellites, Ganymede and Callisto, and designs the
orbital sequence to establish the required Jupiter
orbiting conditions required by JMO.
In general, the highly efficient swing-by uses inner
moons. But in such orbits, the spacecraft is exposed
to the strong radiation environment of Jovian system.
The strategy to relieve the effect of radiation is taken
into account in the orbital sequence design. Also the
plane change transfer without using deterministic
delta-V is discussed.
2. Basic Properties of Multiple Swing-by
Trajectory
2.1. Basic Consideration of Jovian Moons
Swing-by
Table 1 provides a summary of important physical
and orbital data for each of the Galilean satellites.
These satellites are all in near-ecliptic orbits, and the
inner three satellites' orbital periods are locked in
1:2:4 resonance. There are more than 12 additional
satellites, all of which are much smaller than the
Galilean satellites. Because these satellites have very
low masses, their gravitational influence on the
spacecraft's trajectory is negligible. These smaller
satellites are, therefore, not used for orbit transfers.
Ganymede, the most massive satellite, is effective
for changing orbital period and inclination. It is
particularly effective at accomplishing large
reductions in orbital period.
Callisto, the outermost Galilean satellite, is less
massive than Ganymede and orbits at a greater
distance from Jupiter. Therefore, Callisto is less
effective than Ganymede in changing orbital period.
However, because Callisto orbits further from
Jupiter than Ganymede, a given period change at
Callisto results in a greater change in perijove
distance and a greater apsidal rotation than at
Ganymede. Also, the smaller radiation effect is
expected than Ganymede flyby. Although not as
massive as Ganymede, it can be more effective in
changing inclination because the Orbiter's velocity is
lower at the greater distance from Jupiter at which
Callisto is encountered.
Europa, the least massive of the Galilean satellites,
is least effective in changing orbital period and
inclination. Since the Orbiter's perijove cannot be
much closer to Jupiter than the distance at which
Europa orbits due to radiation considerations,
targeted flybys with Europa must occur close to
perijove. Therefore, the amount of change in
perijove distance and rotation of the line of apsides
caused by energy changes at Europa is small.
Consequently, in this paper, the resonance orbits of
Ganymede and Callisto are synthesized. The
patched-conics analysis is performed, in which the
orbits are connected one by the next by satisfying
the swing-by conditions at Ganymede and Callisto.
2.2. Parameters Change Before and After
Swing-by
The multiple swing-by technique uses the orbits
which have the integer ratio of that of a Jovian moon.
In this paper, such resonance orbits are utilized. The
spacecraft is supposed to transfer from one
resonance orbit to another, satisfying swing-by
conditions.
Flybys that change the orbital period also rotate the
line of apsides and change the perijove distance. For
a given orbital period change, a flyby that occurs far
from the spacecraft's perijove rotates the line of
apsides and changes perijove distance more than a
flyby occurring close to perijove. A flyby occurring
exactly at perijove changes orbital period with
minimal changes to the line of apsides and perijove
distance. The direction in which the line of apsides is
rotated depends on whether the orbital period is
increased or decreased and whether the satellite
flyby occurs before perijove (ginboundh) or after
perijove (goutboundh). The inbound flyby rotates the
apsides counter-clockwise for energy reducing flyby,
while the outbound flyby rotates the apsides
clockwise for energy reducing flyby. These
characteristics can be used to control the shape and
the direction of the spacecraftfs orbit without using
the active orbit transfer maneuver using RCS.
Consider hereafter the coplanar energy reducing
orbit transfer. Denoting orbital energy, angular
momentum, velocity, radius, gravity constant and
inplane flight path angle as E,h,v,r,,, then the
following relations are satisfied;
1 2
2
E v
r
= ? (1)
h=rvsin (2)
Table 1: Galilean Satellite Physical and Orbital Data [1]
2 2 2
2 2
2 2
2 sin e1 Eh 1 r v2 v
r
= + = + ?? ? ??
? ?
(3)
Suppose the change in the inplane flight path angle
at flyby with reference to the central body (Jupiter)
is very small. This assumption is valid when the
orbit energy is sufficiently high compared with that
of the moon to be used for swing-by. Then the
change in the eccentricity can be written as follows;
e4v r2sin2 1 1 v
e r a
= ?? ???????
? ? ??
(4)
where a is the semimajor axis. Because the flyby
occurs at the perijove side, r > ??
?< <
(9)
The inequations (9) support the aforementioned
relations; the inbound flyby rotates the apsides
counter-clockwise, and the outbound flyby rotates
the apsides clockwise.
From the relation between perijove radius rp and e,.
and from equation (9),
( )
( )
( ) p 2 2
1 cos sin 1 2 cos
1 1 1
r e e e e
e e e
ƃ ƃ
=? + ? = ? + +
+ + +
(10)
is obtained. (10) implies that the any flybys
performed at -60<<60deg reduces the perijove
radius, which is the general case of the Galilean
satellites high energy flybys.
In summary, it can be said that the energy reducing
high-energy flybys
? always reduce the eccentricity (from eq (5))
? rotate the apsides (from eq.(9))
? always reduce the perijove radius (from eq.(10))
3. Jupiter Tour Trajectory Design
3.1. Ganymede Swing-by Trajectory
The patched-conics analyses are performed to obtain
the multiple swing-by trajectories. The initial orbit
must be a resonance orbit with the moon to be used
for the first swing-by. Therefore, at Jupiter orbit
insertion (JOI) from the interplanetary orbit, the
spacecraft is assumed to be inserted to the 1:50
Ganymede resonance orbit or 1:22 Callisto
resonance orbit, both of which have the orbital
periods of about 360 days.
The requirements for the final (mission) orbit of
JMO are as follows;
? Perijove is 15 Rj (1Rj=71492km). The time to
stay within 13 Rj distant from Jupiter should be
short to avoid the severe radiation environment.
? Apojove is around 50 to 80 Rj.
? The orbital plane is near ecliptic, and can be
slightly inclined to avoid long eclipse if needed.
This subsection exclusively uses Ganymede for
orbit transfer. In this preliminary study, the
following conditions are imposed to obtain the
Jupiter tour trajectory.
? The initial orbit is 1:50 Ganymede resonance
orbit with 13 Rj perijove, and the orbital plane
aligns with the ecliptic plane.
? The epoch for the first swing-by is MJD=58849.
? The final orbit is 1:3 Ganymede resonance orbit.
? The total flight time from JOI to the final
swing-by must be less than 800days.
? The minimum swing-by altitude limitation is
500km.
? Only the orbital sequences which require no
deterministic delta-V is considered.
All the possible orbital sequences are searched and
consequently, 10 solutions are obtained (Table 2),
which are all composed of 5 orbit nodes (i.e. 3
swing-bys). In Table 2, G50-G12-G05-G03 means
that each orbit is Ganymede-resonant with the
resonance ratio of 50, 12, 5 and 3. The character e-f
means inbound flyby, while e+f means outbound
flyby. The example trajectory is shown for the orbit
sequence (2) (G50-G11-G05-G03) in Figure 1.
Figure 2 provides the orbit properties changes as
results of swing-bys. The orbital energy and the
period smoothly decreases at one swing-by to
anther, and at the same time, the perijove distance
decreases monotonically as is suggested by
equation (10).
To effectively take advantage of Ganymede gravity,
the initial perijove of the spacecraft is sufficiently
small than the 15 Rj Jupiter to Ganymede distance.
The 13 Rj initial perijove in this analysis is selected
by this reason, and that makes it inevitable for the
spacecraft to fly with the considerably low perijove
during the Jupiter tour.
3.2. Callisto Swing-by Trajectory
The same analysis is performed for Callisto
swing-by case.
Figure 3 provides the result analyzed with the
following conditions;
? The initial orbit is 1:22 Callisto resonance orbit
with 13 Rj perijove, and the orbital plane aligns
with the ecliptic plane.
? The final orbit is 1:2 Callisto resonance orbit.
The other conditions are the same as section 3.1.
The resulting sequences require 4 Callisto-flyby, and
the flight time ranges from 630 to 750 days, which is
longer than the Ganymede siwng-by cases.
Because Callisto is further than Ganymede from
Jupiter, the true anomaly at which the swing-by
occurs is larger for Callisto swing-by. It leads to the
larger perijove variation at each swing-by, and
consequently, the final perijove is even worse
(around 7 Rj) compared with the Ganymede
swing-by case.
The merit of using Callisto for swing-by can be
brought out by choosing larger initial perijove.
Figure 5 shows the results of Callisto swing-by
sequences with the initial perijove set to be 23 Rj. In
this case, the flight time ranges from 530 to 690 days,
and the final orbit perjove is 18 Rj, which is
sufficiently high as long as the radiation effect is
concerned.
If one wants to establish the perijove of 23 Rj from
Table 2: Orbit Sequence of Ganymede Multiple
Swing-bys
No. Orbital Sequence Flight Time [day]
(1) G50-G12-G05-G03 500.8
(2) G50-G11-G05-G03 493.7
(3) G50-G10-G05-G03 486.5
(4) G50+G19+G06+G03 558.1
(5) G50+G13+G05+G03 508.0
(6) G50+G12+G05+G03 500.8
(7) G50+G11+G05+G03 493.7
(8) G50+G10+G05+G03 486.5
(9) G50+G09+G05+G03 479.4
(10) G50+G09+G04+G03 479.2
Figure 1: Example of Ganymede Multiple
Swing-by Trajectory (G50-G11-G05-G03)
GanymeCallisto de
Figure 2: Orbit Properties Changes in Ganymede
Multiple Swing-by
(The horizontal axis is the number of swing-by for the right bottom
figure, and the number of orbital node for other three figures)
13 Rj perijove orbit after JOI, 200m/s delta-V
apojove maneuver is required.
3.3. Ganymede/Callisto Swing-by Trajectory
The Ganymede and Callisto combined flyby
sequence is investigated. The solution for this case is
much sensitive to the initial time, because it depends
on the relative phase around Jupiter between
Ganymede and Callisto, so the analysis was done
with a particular initial time. Also it should be noted
that the first priority of this case is the timing to
accomplish flybys, so that the orbital energy may be
increased after flybys.
The conditions for analysis are as follows;
? The initial orbit is 1:50 Ganymede resonance
orbit with 13 Rj perijove, and the orbital plane
aligns with the ecliptic plane.
? The final orbit is 1:2 Callisto resonance orbit.
? One transfer from Ganymede resonance orbit to
Callisto resonance orbit is allowed.
The other conditions are the same as section 3.1.
Figure 7 shows the result of Ganymde/Callisto
swing-by case. The number of swing-by in this case
is 5, and the flight time ranges from 530 to 640 days.
Even though the initial perijove is set to be 13 Rj,
several solutions with larger than 13 Rj perijove are
obtained.
4. Plane Change Strategy in Jupiter System
The ecliptic plane orbit generally has a long eclipse
time, and it is undesirable because it makes the
design of spacecraft system and mission operation
more complicated.
Figure 9 provides the basic characteristics of the
eclipse of the ecliptic plane orbit. The apojove
eclipse does not have to be taken into account,
because practically it cannot occur when considering
the JOI direction condition. The other cases can
occur during Jupiter tour. This is inevitable because
the Jupiterfs moon multiple swing-by impose the
spacecraftfs orbital plane to be aligned to the ecliptic
plane (Galilean satellitesf orbital plane).
Figure 3: Orbit Properties Changes in
Callisto Multiple Swing-by (Rp=13Rj)
Figure 5: Orbit Properties Changes in
Callisto Multiple Swing-by (Rp=23Rj)
Figure 4: Example Trajectory for Callisto
Multiple Swing-by (Rp=13Rj)
(C22-C07-C05-C03-C02)
Figure 6: Example Trajectory for Callisto
Multiple Swing-by (Rp=23Rj)
(C22-C05-C03-C02)
After the Jupiter tour ends, on the other hand, the
orbital plane of JMO is free to be chosen. A small
inclination can reduce the eclipse time drastically.
(Even 0 eclipse time is possible)
The swing-by to change the orbit plane can be done
with one selected Jupiter moon. As mentioned in
section 2.1., Callisto is preferable for this objective.
On the contrary with energy reducing swing-by, the
inclination change swing-by requires the spacecraft
to pass the polar region of Callisto.
The example trajectory is shown in Figure 10. The
two additional swing-bys are added to the example
trajectory provided in Figure 8. As a result of these
additional two swing-bys, 5 degree inclination is
established. Because the ascending and descending
nodes of the finally established orbit are sufficiently
far from the Sun direction, in this case, the eclipse
time for the final orbit is accomplished to be zero.
5. Conclusion
The multiple swing-by trajectory in Jupiter system
was investigated.
This paper focused on flyby with two large Galilean
satellites, Ganymede and Callisto, and designed the
orbital sequence to establish the required Jupiter
orbiting conditions required by Jupiter Magnetsphric
Orbiter (JMO), which had been proposed as the
JAXA/ESA joint mission in Cosmic Vision
framework.
The major results obtained in this study are listed as
follows;
? Ganymede multiple swing-by can establish the
final objective apojove of 50 to 70 Rj, but the final
perijove is lower than required, which makes the
spacecraft to be exposed in the severer radiation
environment.
? Callisto multiple swing-by with the same initial
orbit conditions is inadvisable, because it results in
even lower perijove than Ganymede swing-by
trajectories.
Figure 7: Orbit Properties Changes in
Ganymede/Callisto Multiple Swing-by
Figure 8: Example Trajectory for
Ganymede/Callisto Multiple Swing-by
(G50+G20+G06+C-C02)
(9.8hr)
5.8hr
6.1hr
--- (very long)
Apojovian Eclipse
22.9days
11.9days
10.9days
232.6days
Orbital Period
4.5hr
4.2hr
3.9hr
2.9hr
=90deg Eclipse
3.2h15RJ x 27RJ r
15RJ x 50RJ 3.0hr
12.7RJ x 27RJ 2.9hr
5RJ x 300RJ 1.5hr
Orbit Perijovian Eclipse
15RJ 3.2hr
Apojovian Eclipse
Perijovian Eclipse
=90deg Eclipse
Jupiter
Spacecraft Trajectory
Figure 9: Basic Eclipse Characteristics of
Ecliptic Plane Orbit around Jupiter
Figure 10: Example Jupiter Tour with 5
deg Inclination Final Orbit
(G50+G20+G06+C-C02-C02-C02)
? Callisto multiple swingby can establish the final
objective apojove and perijove, when the initial
orbit perijove after JOI is tuned to be as high as
around 23 Rj.
? Ganymede and Callisto combined multiple
swing-by broadens the possible orbital sequence
solutions, although it is more dependent on the JOI
epoch.
? Additional a few Callisto swing-bys sufficiently
provide the inclination change to reduce the
eclipse time.
Reference
[1] Louis A. Dfamario, Larry E. Bright and Aron A.
Wolf, gGalileo Trajectory Designh, Space
Science Reviews, vol. 60, no. 1-4, May 1992, pp.
23-78.