ORBIT DESIGN AND CONTROL OF PLANETARY SATELLITE ORBITERS IN THE HILL 3-BODY PROBLEM by Marci Paskowitz Possner A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Aerospace Science) in The University of Michigan 2007 Doctoral Committee: Associate Professor Daniel J. Scheeres, Chair Professor Pierre T. Kabamba Professor N. Harris McClamroch Associate Professor Thomas Zurbuchen
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ORBIT DESIGN AND CONTROL OF
PLANETARY SATELLITE ORBITERS IN THE
HILL 3-BODY PROBLEM
by
Marci Paskowitz Possner
A dissertation submitted in partial fulfillmentof the requirements for the degree of
Doctor of Philosophy(Aerospace Science)
in The University of Michigan2007
Doctoral Committee:
Associate Professor Daniel J. Scheeres, ChairProfessor Pierre T. KabambaProfessor N. Harris McClamrochAssociate Professor Thomas Zurbuchen
2. OVERVIEW OF THE DYNAMICAL SYSTEM . . . . . . . . 92.1 The Three Body and Restricted Three Body Problems . . . . 102.2 The Hill 3-Body Problem . . . . . . . . . . . . . . . . . . . . 132.3 The Modified Hill 3-Body Problem . . . . . . . . . . . . . . . 18
5. THE DESIGN OF LONG LIFETIME SCIENCE ORBITS . 935.1 Dynamics in the 1-DOF System . . . . . . . . . . . . . . . . 955.2 Computing Initial Conditions in the 3-DOF System . . . . . . 1005.3 Long Lifetime Orbits in the 3-DOF System . . . . . . . . . . 1095.4 A Toolbox for Computing Long Lifetime Orbits . . . . . . . . 120
6. CONTROL OF LONG LIFETIME ORBITS . . . . . . . . . . 122
2.1 Schematic of the Circular Restricted 3-Body Problem . . . . . . . . 122.2 Schematic of the Hill 3-Body Problem . . . . . . . . . . . . . . . . . 142.3 Trajectory integration in the Modified Hill 3-body problem (Third
degree and order gravity field) . . . . . . . . . . . . . . . . . . . . . 222.4 Trajectory integration using the full ephemeris model, and Europa’s
gravity field up to third degree and order [3]. . . . . . . . . . . . . . 232.5 Trajectory integration in the Modified Hill 3-body problem (truncated
3.1 A frozen orbit in the 1-DOF system integrated over several orbitalperiods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Frozen orbit solutions for the tide-only case. . . . . . . . . . . . . . 403.3 Comparison of frozen orbit solutions with and without J2. . . . . . . 423.4 Frozen orbit solutions for J2 6= 0 and J3 > 0. . . . . . . . . . . . . . 433.5 Frozen orbit solutions with larger and smaller values of J3. . . . . . 453.6 Stability of frozen orbits about Europa for the tide-only case. . . . . 473.7 Stability of the frozen orbits for the tide plus J2 case. . . . . . . . . 483.8 Stability for frozen orbits about Europa that include the tidal term,
J2 and J3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.9 Characteristic times for the tide-only unstable frozen orbits with 100
km altitudes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.10 Characteristic times for unstable frozen orbits about Europa . . . . 513.11 Stability and characteristic times of frozen orbits about Europa with
J3 one order of magnitude smaller than Table 3.1 . . . . . . . . . . . 523.12 Stability and characteristic times of frozen orbits about Europa with
J3 one order of magnitude larger than Table 3.1 . . . . . . . . . . . 533.13 Contour plots for motion in the vicinity of a stable frozen orbit with
e∗ = 0.548, i∗ = 50o, J3 sin ω > 0. . . . . . . . . . . . . . . . . . . . . 553.14 Contour plots for motion in the vicinity of an unstable frozen orbit
with e∗ = 0.0.039, i∗ = 55o, J3 sin ω > 0. . . . . . . . . . . . . . . . . 563.15 Contour plots for motion in the vicinity of a stable frozen orbit with
e∗ = 0.047, i∗ = 30o, J3 sin ω < 0. . . . . . . . . . . . . . . . . . . . . 583.16 Contour plots for motion in the vicinity of an unstable frozen orbit
5.3 Stable and unstable manifolds for a frozen orbit, identified with an’x’, with e∗ = 0.0129, i∗ = 70o, ω∗ = −90o and a∗ = 1682.5 km. . . . 98
5.4 Expansion of the central region in 5.3(b) with Europa impact circle . 995.5 3-DOF system integration of a frozen orbit. . . . . . . . . . . . . . . 1005.6 Integration of trajectory initialized on 1-DOF manifold. . . . . . . . 1025.7 Integration of trajectories initialized at xm
5.8 Integration of trajectories initialized at x0+δx0 in 3-DOF system andx0 in 2-DOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.9 Integration in the 3-DOF of a trajectory initialized at x0 with the1-DOF manifolds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.10 Time history of the eccentricity for a long lifetime orbit. . . . . . . . 1115.11 1-DOF manifolds with minimum eccentricity circle (dashed curve). . 1125.12 Long lifetime orbits about Europa (70o frozen orbit inclination) . . . 117
x
5.13 Long lifetime orbit about Enceladus (70o frozen orbit inclination) . . 1195.14 Long lifetime orbit about Dione (70o frozen orbit inclination) . . . . 120
6.1 Time histories of orbital elements integrated in the controlled 1-DOFsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2 Time histories of orbital elements integrated in the uncontrolled 1-DOF system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.3 Integration of the 3-DOF system controlled with Fr. . . . . . . . . . 1296.4 Manifold in the 1-DOF system and integration of the 3-DOF system
controlled with Fr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.5 Time histories of total thrust magnitude and components of thrust. . 1316.6 Total ∆v used to control the spacecraft using the thrust law given in
Eq.(6.13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.7 Integration using constant thrust in the radial direction of −9.25×10−7.1326.8 Integration using constant thrust in the radial direction of−1.045×10−6.1336.9 Eccentricity as a function of time over one orbital period. . . . . . . 1346.10 Time histories of the orbital elements when the transverse thrust is
applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.11 Diagram of the reset scheme in the 1-DOF system in (e,ω)-space. . . 1376.12 Original (black) and target (red) long lifetime trajectories. . . . . . . 1446.13 Time histories of the orbital elements of the original (black) and target
(red) long lifetime trajectories. . . . . . . . . . . . . . . . . . . . . . 1446.14 Monte Carlo simulation results for various long lifetime trajectories. 1486.15 Schematic of Hohmann transfer between two elliptic orbits. . . . . . 151
4.1 Comparison of costs to transfer from a capture trajectory to a circularfrozen orbit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Comparison of costs to transfer from capture trajectory to tightlybound circular orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.1 Algorithm for computing a long lifetime orbit . . . . . . . . . . . . . 1145.2 Long Lifetime Orbits for the Europa System . . . . . . . . . . . . . 1165.3 Long Lifetime Orbits about Enceladus . . . . . . . . . . . . . . . . . 1185.4 Long Lifetime Orbits about Dione . . . . . . . . . . . . . . . . . . . 118
6.1 Examples of Long Lifetime Orbit Reset . . . . . . . . . . . . . . . . 1436.2 Monte Carlo Results for Long Lifetime Orbits . . . . . . . . . . . . . 1476.3 Average cost to correct a long lifetime trajectory after an initial po-
sition error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1546.4 Average cost to correct a long lifetime trajectory after a larger initial
Europa orbit rate N 2.05× 10−5 rad/sEuropa orbital period TE 3.55 daysEuropa eccentricity es 0.01Europa semi-major axis as 671100 kmEuropa radius Rs 1560.8 kmEuropa J2 J2 1041.39 km2
Europa C21 C21 −0.32 km2
Europa S21 S21 −20.47 km2
Europa C22 C22 312.97 km2
Europa S22 S22 −4.71 km2
Europa J3 J3 524117.32 km3
Europa C31 C31 120230.75 km3
Europa S31 S31 81414.49 km3
Europa C32 C32 −11938.14 km3
Europa S32 S32 −3617.07 km3
Europa C33 C33 −1765.93 km3
Europa S33 S33 3669.41 km3
21
0 20 40 60 80 100 1201550
1600
1650
1700
1750
1800ra
dius
(km
)
time (days)
0 20 40 60 80 100 1201660
1661
1662
1663
1664
1665
1666
1667
1668
sem
i−m
ajor
axi
s (k
m)
time (days)
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ecce
ntric
ity
time (days) 0 20 40 60 80 100 120
109.5
110
110.5
111
111.5
112
incl
inat
ion
(deg
)
time (days)
0 20 40 60 80 100 120−120
−100
−80
−60
−40
−20
0
20
40
ω (
deg)
time (days)
0 20 40 60 80 100 120−200
−150
−100
−50
0
50
100
150
200
Ω (
deg)
time (days)
Figure 2.3: Trajectory integration in the Modified Hill 3-body problem (Third degree and ordergravity field)
22
Figure 2.4: Trajectory integration using the full ephemeris model, and Europa’s gravity field up tothird degree and order [3].
23
where the values of J2, C22 and J3 have been dimensionalized by their multiplications
by R2s and R3
s, respectively. Therefore, we verify that including only the J2, C22
and J3 coefficients does not drastically affect the results. In a real mission design
situation, analysis would be performed using this reduced gravity field and a
trajectory valid in the Hill problem would be computed. This trajectory could then
be used as a starting point for designing a trajectory in the full ephemeris model.
In examining results obtained using a truncated Europa gravity field, it is not
necessary to find the exact same behavior of the trajectory, especially in terms of
its lifetime. The environment around Europa in particular is highly perturbed, and
a slight change in the initial conditions can result in a drastic change in the overall
trajectory. Therefore, we are only interested in verifying that using a model with a
truncated gravity field captures the important features of the motion. Comparing
Figure 2.5 which shows the integration of a trajectory in the modified Hill 3-body
problem with the gravity field truncated to J2, C22 and J3 to Figure 2.3 which
shows an integration using the third degree and order gravity field, we see that the
overall features are very similar. Specifically, the amplitudes of the oscillations in
inclination and semi-major axis are the same in both cases, as are the trends in the
motion of all of the orbital elements. The main difference between the two cases
is the amplitude of the eccentricity and argument of periapsis oscillations. In the
third degree and order gravity field case, the size of the oscillations are larger for
both orbital elements. However, since the average motion remains the same between
the full gravity field and truncated gravity field case, the results obtained from the
truncated gravity field case are sufficient for our purposes.
2.3.2 Planetary Satellite Eccentricity
In this section the assumption that the planetary satellite is in a circular orbit about
the planet is verified. To do so, an elliptic planetary satellite orbit is added to the
24
0 20 40 60 80 100 1201550
1600
1650
1700
1750
1800ra
dius
(km
)
time (days)0 20 40 60 80 100 120
1661
1662
1663
1664
1665
1666
1667
1668
sem
i−m
ajor
axi
s (k
m)
time (days)
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ecce
ntric
ity
time (days)0 20 40 60 80 100 120
109.8
110
110.2
110.4
110.6
110.8
111
111.2
111.4
111.6
111.8
incl
inat
ion
(deg
)
time (days)
0 20 40 60 80 100 120−100
−80
−60
−40
−20
0
20
40
ω (
deg)
time (days)0 20 40 60 80 100 120
−200
−150
−100
−50
0
50
100
150
200
Ω (
deg)
time (days)
Figure 2.5: Trajectory integration in the Hill 3-body problem, with the Europa gravity fieldtruncated to include only the J2, C22 and J3 contributions.
25
model. In the circular model, the assumption that the rotation rate of the planetary
satellite is synchronous with its orbit rate was made. Therefore, when an elliptic
planetary satellite orbit is introduced, it is important to ensure that the rotation
rate of the planetary satellite is still constant, and does not vary as the speed of
the planetary satellite on its orbit varies. For these comparisons, the modified
Hill 3-body problem includes the J2, C22 and J3 planetary satellite gravity field
components.
The equations of motion for the elliptic Hill 3-body problem are most easily
expressed in non-dimensional coordinates. The potential of the non-dimensional
circular Hill problem is:
VC =1
r+
1
2(x2 + y2) + RC , (2.54)
where
RC = −1
2r2+
3
2x2− J2
2ε2d2
(3z2
r5− 1
r3
)+
3C22
ε2d2r5(x2−y2)− J3
2ε2d2
(5z3
r7− 3z
r5
). (2.55)
The factors used to remove the dimensions of the system are ε and the radius of the
planetary satellite in orbit about the planet, d:
ε = (µ/µp)1/3 , (2.56)
d =as(1− e2
s)
1 + es cos fs
, (2.57)
where µp is the gravitational parameter of the planet, as is the semi-major axis
of the planetary satellite, es is the eccentricity of the planetary satellite and fs is
the true anomaly of the planetary satellite. Then, the potential of the elliptic Hill
3-body problem is[13]:
VE =1
1 + es cos fs
VC . (2.58)
One final correction to the potential must be made, to ensure the constant rotation
rate of the planetary satellite. This correction only applies to the C22 component of
26
the potential since it is the only non-symmetric component. Specifically, the x and
y values in the C22 component in Eq.(2.55) are replaced by x′ and y′, where x′
y′
=
cos (Ms − fs) sin (Ms − fs)
sin (Ms − fs) cos (Ms − fs)
x
y
, (2.59)
and Ms is the mean anomaly of the planetary satellite. Then, the equations of
motion for the elliptic modified Hill 3-body problem are:
x = 2y +∂VEH
∂x, (2.60)
y = −2x +∂VEH
∂y, (2.61)
z =∂VEH
∂z, (2.62)
where
VEH =1
1 + es cos fs
[1
r+
1
2(x2 + y2) + REH
], (2.63)
and
REH = −1
2r2 +
3
2x2 − J2
2ε2d2
(3z2
r5− 1
r3
)+
3C22
ε2d2r5(x′
2 − y′2)
− J3
2ε2d2
(5z3
r7− 3z
r5
). (2.64)
Figure 2.6 shows the results of the integration of a trajectory in the elliptic
modified Hill 3-body problem. The parameters used are those for a spacecraft in
orbit about Europa, and are given in Table 2.1. Compared to Figure 2.5, it is clear
that the main features of the motion are the same. The orbital elements show the
same trends, and the sizes of the oscillations are the same. The major difference
is that in the elliptic case, the semi-major axis shows a slow decrease. This is
not a large concern, since the overall decrease is only about 2 km over 130 days.
Ignoring this effect in the circular case will not strongly affect the overall analysis.
Therefore, for ease of analysis, the Circular Modified Hill 3-body problem is used for
27
the remainder of this dissertation.
28
0 20 40 60 80 100 120 1401560
1580
1600
1620
1640
1660
1680
1700
1720
1740
1760ra
dius
(km
)
time (days)0 20 40 60 80 100 120 140
1656
1658
1660
1662
1664
1666
1668
sem
i−m
ajor
axi
s (k
m)
time (days)
0 20 40 60 80 100 120 1400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
ecce
ntric
ity
time (days)0 20 40 60 80 100 120 140
109
109.5
110
110.5
111
111.5
incl
inat
ion
(deg
)
time (days)
0 20 40 60 80 100 120 140−120
−100
−80
−60
−40
−20
0
20
40
ω (
deg)
time (days)0 20 40 60 80 100 120 140
−60
−40
−20
0
20
40
60
80
100
120
140
Ω (
deg)
time (days)
Figure 2.6: Trajectory integration in the Elliptic Modified Hill 3-body problem, with the Europagravity field truncated to include only the J2, C22 and J3 contributions.
29
CHAPTER 3
FROZEN ORBITS
The Hill 3-body problem is very complex in that it is a 3 degree of freedom (DOF)
non-integrable system. Detailed analysis of it in that form is only possible through
numerical simulations and it is difficult to obtain insight into the system except for
some special situations such as equilibrium points. Given that the ultimate goal of
the study of planetary satellite orbiters is the design of useful orbits for a mission,
it is important to learn more than what can be gained from numerical simulations.
This is accomplished with the analysis of a reduced system. By averaging over the
motion of the spacecraft about the planetary satellite and over the motion of the
planetary satellite about the planet, the 3-DOF system can be reduced to a 1-DOF
integrable system. This 1-DOF system has equilibrium solutions, denoted as frozen
orbits, which have constant values of the eccentricity, inclination and argument of
periapsis, on average. We compute frozen orbit solutions for three models: tide-only,
tide + J2 and tide + J2 + J3 and characterize the differences between them.
Previous work using averaging methods has studied the dynamics of orbits
about planetary satellites and frozen orbits. Broucke [7] studied orbit dynamics in
a model that included the tidal force. He found elliptic and circular frozen orbits
and investigated their stability. Scheeres, et al. [37] studied near-circular frozen
orbits in a model that included both the tidal force and J2. They investigated the
stability of these orbits and developed an analytic theory to postpone impact with
the surface for significant periods of time. In addition San-Juan et al. [34] and Lara
30
et al. [22] studied orbit dynamics about planetary satellites using a more rigorous
averaging method that involved Lie transforms to develop averaged equations to a
higher order. They included the tidal term and J2 and the tidal term, J2 and C22,
respectively.
Aside from computing the frozen orbit solutions, we also examine their stability
since some frozen orbits are stable and others are unstable. The concept of the
characteristic time is also introduced. Just as the distribution of the frozen orbit
solutions in orbit element space is affected by the inclusion of the gravity field
coefficients, so too is their stability. The stability of the frozen orbit solutions is
therefore computed for the three different cases.
Contour plots are a way to study the qualitative secular motion in the vicinity
of frozen orbits. The motion in the vicinity of the different types of frozen orbits
is shown using contour plots in the 1-DOF system and integrations in the 3-DOF
system. The integration of frozen orbit solutions in the 3-DOF system is used to
determine whether the stable frozen orbits are viable as trajectories in the 3-DOF
system and to show that unstable frozen orbits in the 1-DOF system are also
unstable in the 3-DOF system.
3.1 Reduction of the System by Averaging
Recall the perturbing potential of the 3-DOF system:
R =1
2N2(3x2 − r2
)− µJ2
2
(3z2
r5− 1
r3
)+
3µC22
r5(x2−y2)− µJ3
2
(5z3
r7− 3z
r5
). (3.1)
Before presenting the results of the reduction of the system, it is necessary to
introduce three regimes other than Cartesian coordinates in which the motion can
be analyzed. Orbital elements are a set of six values used frequently in celestial
mechanics and astrodynamics to describe the size, shape and orientation of an orbit.
Semi-equinoctial elements are useful for dealing with near-circular orbits. Delaunay
31
variables are a canonical set of six elements of which three are angles and three are
momenta. The equations of motion in terms of Delaunay variables are quite simple
and lead directly to the idea of the reduction of the system by averaging.
The orbital elements uniquely define a Keplerian orbit, given the Cartesian
position and velocity. For the case of the motion of a point mass in a central
gravitational field (which is integrable) they describe the specific conic section that
the path of the particle follows. In our case, that of a spacecraft in a non-central
field, the orbital elements vary as functions of time and represent a convenient way
to describe the motion. The orbits that we are primarily interested in are elliptical.
The six orbital elements most commonly used are:
• a: semi-major axis
• e: eccentricity
• i: inclination
• ω: argument of periapsis
• Ω: longitude of the ascending node
• M : mean anomaly
Some other orbital elements that we use are:
• ra = a(1 + e): radius of apoapsis
• rp = a(1− e): radius of periapsis
• E: eccentric anomaly (M = E − e sin E)
• f : true anomaly(tan f/2 =
√1+e1−e
tan E/2)
32
All of the above orbital elements are in the inertial frame. Since the motion that we
consider takes place in the rotating frame, the longitude of the ascending node in
the rotating frame is also used:
Ω = Ω−N(t− t0) , (3.2)
where t0 is an initial epoch.
The equations of motion in terms of the orbital elements are obtained from
the Lagrange Planetary Equations, where R is the perturbing potential under
consideration[11]:
da
dt=
2
na
∂R
∂M, (3.3)
de
dt=
1− e2
na2e
∂R
∂M−√
1− e2
na2e
∂R
∂e, (3.4)
di
dt=
1
na2√
1− e2
[cot i
∂R
∂ω− csc i
∂R
∂Ω
], (3.5)
dω
dt=
√1− e2
na2e
∂R
∂e− cot i
na2√
1− e2
∂R
∂i, (3.6)
dΩ
dt=
1
na2√
1− e2 sin i
∂R
∂i, (3.7)
dM
dt= n− 1− e2
na2e
∂R
∂e− 2
na
∂R
∂a. (3.8)
Semi-equinoctial elements, denoted as h and k, are defined as [24]:
h = e sin ω ,
k = e cos ω .
This set of variables is particularly useful for near-circular orbits since all of the
motion is concentrated in a small region of possible (h, k) space, making for easier
analysis and plotting.
The Delaunay variables consist of the three angles M , ω and Ω complemented by
33
three corresponding momenta, denoted as L, G and H and defined as follows:
L =√
µa ,
G = L√
1− e2 , (3.9)
H = G cos i .
The equations of motion of the 3-DOF system in terms of Delaunay variables are[8]:
dL
dt=
∂R
∂M,
dM
dt= n− ∂R
∂L,
dG
dt=
∂R
∂ω,
dω
dt= −∂R
∂G, (3.10)
dH
dt=
∂R
∂Ω,
dΩ
dt= − ∂R
∂H.
The fact that the full system has three degrees of freedom can be seen from the
Delaunay variable representation. The three degrees of freedom correspond to the
three free angular coordinates, M , ω and Ω. Note that H is the z-component of
the angular momentum. The Jacobi integral, defined in Eq.(2.46) in Cartesian
coordinates can be expressed in osculating orbital elements as:
J = − µ
2a−NH −R(a, e, i, ω, Ω, M) . (3.11)
Averaging is an approximation technique that can be used to remove degrees of
freedom from this system. In general, averaging occurs over an angle of the system
and causes its corresponding momentum to be conserved. For the first averaging, we
assume that the spacecraft’s mean motion about the planetary satellite, n =√
µ/a3,
is much greater than N , the mean motion of the planetary satellite about the
planet[37]. This means that over the time it takes for one revolution of the spacecraft
about the planetary satellite, the planetary satellite will only have moved by a small
amount around the planet. For a low-altitude orbit about Europa, the ratio of N/n
is ∼ 0.02− 0.03 which means that Europa will only have moved around Jupiter by
34
about 7-10 degrees. This condition is also satisfied by many other planetary satellites
in our solar system. Two in particular to be discussed later are Saturn’s moons
Enceladus and Dione. They have N/n ratios of ∼ 0.09 and ∼ 0.04, respecively [37].
The first averaging, to reduce the system to 2 degrees of freedom is over the
mean anomaly of the spacecraft about Europa; define
R =1
2π
∫ 2π
0
RdM . (3.12)
The computation of R requires the definition of the Cartesian coordinates x, y and
z as:
x = r[cos (ω + ν) cos Ω− sin (ω + ν) sin Ω cos i] , (3.13)
y = r[cos (ω + ν) sin Ω + sin (ω + ν) cos Ω cos i] , (3.14)
z = r sin i sin (ω + ν) , (3.15)
where t is the time. Note that for this computation, Ω is assumed to be constant.
The singly-averaged potential is found to be:
R =N2a2
4
(1− 3
2sin2 i +
3
2cos 2Ω sin2 i
)(1 +
3
2e2
)+
15
4e2 cos 2ω
[sin2 i + cos 2Ω
(1 + cos2 i
)]− 15
2e2 sin 2ω sin 2Ω cos i
+
µJ2
2a3(1− e2)3/2
(1− 3
2sin2 i
)+
3µC22
2a3(1− e2)3/2cos 2Ω sin2 i
+3µJ3
2a4(1− e2)5/2e sin ω sin i
(1− 5
4sin2 i
). (3.16)
Since we averaged over the mean anomaly, it is no longer present in the potential
defined by Eq.(3.16). Additionally, recall the equation for the time-derivative of
the semi-major axis in Eq.(3.3). Since M does not appear in R, a = 0 and so the
semi-major axis is constant and acts as another integral of motion. Since L =√
µa,
L is also constant. We have reduced the system to a 2-DOF system consisting of the
pairs (H, Ω) and (G, ω).
35
The 2-DOF system is a time-periodic function of Ω. This makes initial analysis
of the qualitative behavior of the system difficult and so we reduce it further by
performing another averaging, over the orbit of the planetary satellite about the
planet. This averaging is justified for Europa since there are order of magnitude
differences between the period of a Europa orbiter (∼ 2 hours) and the period of
Europa about Jupiter (3.5 days) and between the period of Europa about Jupiter
and the time period of the instability of the spacecraft (∼ 1 month)[37]. It is
similarly justified for Enceladus and Dione.
Since Ω is uniformly decreasing with time due to its definition with respect to
the rotating coordinate frame, it can be used as an independent variable in place of
the time. Then, the dynamics are averaged over this new independent variable to
find their simplest form; define
R =1
2π
∫ 2π
0
RdΩ . (3.17)
This defines the doubly averaged potential:
R =N2a2
4
[(1− 3
2sin2 i
)(1 +
3
2e2
)+
15
4e2 cos 2ω sin2 i
]+
µJ2
2a3(1− e2)3/2
(1− 3
2sin2 i
)(3.18)
+3µJ3
2a4(1− e2)5/2e sin ω sin i
(1− 5
4sin2 i
).
This potential has only tidal, J2, and J3 components since the C22 component
vanishes under the averaging. Although this means that the effect of C22 is not
included during the analysis of this system, its effect is considered in later chapters
when the averaging is removed. Since Ω is now eliminated, an additional integral of
motion exists for this system, the z-component of the angular momentum defined
by Eq.(3.9). Since a was constant in the 2 degree of freedom system, it is also
constant in this system. This leads to the definition of a simpler functional form of
36
the integral, θ = H2/µa, or [7]:
θ = (1− e2) cos2 i . (3.19)
Therefore, the system governed by the doubly averaged potential in Eq.(3.18) is a
1-DOF system in (G,ω). To simplify later analysis, recall the Jacobi Integral which,
since it was an integral of motion in the full 3-DOF system, is also in integral of
motion here and has the same value as before the averaging since it is a constant.
Therefore, Eq.(3.11) implies that
J = − µ
2a−NH − R(e, i, ω) . (3.20)
Since a and H are integrals of motion in this system, the potential of the 1-DOF
system, R is also an integral of motion for the reduced 1-DOF system. Note that
this is not true for the potentials of the 2 and 3-DOF systems.
3.2 Frozen Orbits: Equilibrium Solutions of the
1-DOF System
The 1-DOF system is completely described by the variables (G,ω). The equations
of motion of these two variables can be obtained from the 1-DOF potential R as
follows:
dG
dt=
∂R
∂ω, (3.21)
dω
dt= −∂R
∂G. (3.22)
Note that since R is an integral of motion in this system, it can be formally reduced
to quadratures. However, for this analysis it is more useful to consider the time
derivatives of the osculating orbital elements e, i and ω, noting that the equations for
e and i can be combined into the equation for G. These equations are obtained by
substituting the 1-DOF potential, Eq.(3.18) into the Lagrange Planetary Equations
37
given in Eqs.(3.3)-(3.8). Before performing a full-blown analysis of the 6 equations,
some general observations can be made to simplify the analysis. First, note that
the 1-DOF potential is independent of M and Ω. This means that the other orbital
elements don’t depend on M and Ω in the 1-DOF system and so the equations for M
and Ω can be excluded from this analysis. Also, recall that the semi-major axis, a, is
constant in the 1-DOF system, and so its equation of motion can also be excluded.
The three remaining equations of motion, those for e, i and ω, are then the only
ones considered in this section:
de
dt=
15N2
8ne√
1− e2 sin2 i sin 2ω − 3J3n
2a3(1− e2)3sin i
(1− 5
4sin2 i
)cos ω , (3.23)
di
dt=− 15N2
16n
e2
√1− e2
sin 2i sin 2ω +3J3n
2a3(1− e2)3e cos i (1
−5
4sin2 i
)cos ω , (3.24)
dω
dt=
3N2
8n√
1− e2
[5 cos2 i− 1 + 5 sin2 i cos 2ω + e2(1− 5 cos 2ω)
]+
3J2n
4a2(1− e2)2
(1− 5
4sin2 i
)+
3J3n
2a3(1− e2)3
sin ω sin i
e(3.25)
·[(
1− 5
4sin2 i
)(1 + 4e2
)− e2
sin2 i
(1− 19
4sin2 i +
15
4sin4 i
)].
Denote the equilibrium solutions of Eqs.(3.23)-(3.25) as frozen orbits. These
orbits are not frozen in space, but have constant values of e, i and ω on average.
Figure 3.1 shows an example of a frozen orbit integrated in the 1-DOF system over
a few orbital periods.
3.2.1 Tide-only Case
Consider first the tide-only case with J2 = J3 = 0 in Eqs.(3.23)-(3.25). First of all,
for circular orbits where e = 0, de/dt = di/dt = 0 automatically and ω is undefined.
Therefore, circular frozen orbits exist in the tide-only case for all inclination values.
Then with e 6= 0, de/dt = 0 and di/dt = 0 for ω = ±π/2, ω = 0 and ω = π. However,
for ω = 0,π the condition necessary to satisfy dω/dt = 0 is e = 1 which corresponds
38
−3000−2000
−10000
10002000
3000
−3000
−2000
−1000
0
1000
2000
3000−3000
−2500
−2000
−1500
−1000
−500
0
500
1000
1500
x (km)y (km)z
(km
)
Figure 3.1: A frozen orbit in the 1-DOF system integrated over several orbital periods.
to a rectilinear orbit. Since we are only interested in elliptic and circular orbits, we
consider only ω = ±π/2. The e and i solutions for the frozen orbits are then given
by:
5
3cos2 i− 1 + e2 = 0 (3.26)
which gives the following relationship between e and i:
e =
√1− 5
3cos2 i (3.27)
Figure 3.2 shows the frozen orbit solutions for the tide-only case. Note that the
elliptic frozen orbits bifurcate from the circular frozen orbits at i ∼ 39.23o and
i ∼ 140.77o.
An important observation of the tide-only frozen orbit solutions is that they
don’t have any dependence on the system parameters or the altitude of the desired
frozen orbit. Therefore, these frozen orbit solutions are valid for any planetary
satellite/planet system where only the tidal force is considered and for any altitude
frozen orbit.
39
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
Figure 3.2: Frozen orbit solutions for the tide-only case (ω = ±π/2 for elliptic orbits).
3.2.2 Tide and J2 Case
Next, we remove the J2 = 0 condition and evaluate the frozen orbit solutions. Once
again, circular frozen orbits exist for all values of the inclination and elliptic frozen
orbits have the ω = ±π/2 condition. The equation to be solved to obtain the
relationship between eccentricity and inclination for the frozen orbit solutions is
Eq.(3.25) set equal to zero with J3 = 0 and ω = ±π/2:
3N2
3n√
1− e2[10 cos2 i− 6(1− e2)
3nJ2
4a2(1− e2)2
(1− 5
4sin2 i
)= 0 . (3.28)
This yields the following analytic relationship between e and i for the frozen orbit
solutions:
sin2 i =2N2(4 + 6e2)a2(1− e2)3/2 + 2n2J2
20N2a2(1− e2)3/2 + 5n2J2
. (3.29)
One difference between this result and the tide-only result is that the semi-major
axis appears in this solution. In the tide-only case, the frozen orbit solutions were
independent of the semi-major axis, meaning that given either an inclination or
eccentricity, all sizes of orbits would have the same frozen orbit characteristics. In
this case, however, the frozen orbit characteristics will vary as the size of the orbit
40
Table 3.1: Parameters of Europa
Parameter Symbol Value
Europa radius RE 1560.8 km
Europa orbital period T 3.55 days
Europa orbit rate N 2.05× 10−5 rad/s
Europa gravitational parameter µ 3.201× 103 km3/s2
Non-dimensional Europa J2a J2 4.2749× 10−4
Non-dimensional Europa C22a C22 1.2847× 10−4
Non-dimensional Europa J3a J3 1.3784× 10−4
a Values obtained from John Aiello, personal communication, August2004.
varies. Since we are primarily interested in low-altitude orbits, when solving for the
frozen orbit solutions we specify a radius of periapsis and scale the semi-major axis
correspondingly, where:
a =rp
1− e. (3.30)
A periapsis altitude of 100 km is chosen and the values of the other parameters used
are given in Table 3.1.
Figure 3.3 shows the frozen orbit solutions for the tide plus J2 case. The dashed
lines represent the frozen orbit solutions for the tide plus J2 case that differ from the
tide-only case. For eccentricities larger than about 0.4 and circular orbits, the frozen
orbit solutions are not affected by the inclusion of the J2 gravity field coefficient.
However, for elliptic orbits with eccentricities smaller than about 0.4, the frozen
orbit solutions for the tide plus J2 case have larger inclinations than the solutions
for the tide-only case. This occurs because the elliptic frozen orbits bifurcate from
the circular frozen orbits at a larger inclination for the tide plus J2 case than for
the tide-only case. The amount of the inclination difference depends on the value
41
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
tide−onlytide+J
2
Figure 3.3: Comparison of frozen orbit solutions with and without J2 (ω = ±π/2 for elliptic orbits).
of J2 and the desired radius of periapsis. The results shown here are obviously for
particular values of J2 and rp. The overall importance of this result is that for
near-circular frozen orbits, including J2 leads to orbits with higher inclinations which
could be useful for the design of a science orbit about a planetary satellite since in
general, near-polar orbits are desirable for science orbits.
3.2.3 Tide, J2 and J3 Case
Consider now Eqs.(3.23)-(3.25) with J2 6= 0 and J3 6= 0. Again, Eqs.(3.23) and
(3.24) are in equilibrium for ω = ±π/2. Substituting this into Eq.(3.25) and setting
it equal to zero, we obtain:
0 =3
8
N2
n
1√1− e2
[4− 10 sin2 i + 6e2] +3nJ2
4a2(1− e2)2
(1− 5
4sin2 i
)± 3J3n
2a3(1− e2)3
sin i
e
[(1− 5
4sin2 i
)(1 + 4e2
)− e2
sin2 i
(1− 19
4sin2 i +
15
4sin4 i
)], (3.31)
where the ± on the J3 terms corresponds to ω = ±π/2. This implies that the
inclusion of the J3 gravity field coefficient breaks the symmetry between the
ω = ±π/2 frozen orbit solutions. In its place, a symmetry between the signs of J3
42
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
Figure 3.4: Frozen orbit solutions for J2 6= 0, J3 > 0 and ω = ±π/2.
and ω is created, where changing the sign of J3 is equivalent to shifting ω by π. This
is useful in terms of mission design since at this time only the degree two terms of
the Europa gravity field are known [4]. An accurate value of J3 for Europa is not
currently available, including its sign. Due to the existence of the symmetry between
the signs of J3 and ω any desired frozen orbit can be attained, regardless of the sign
of J3 by adjusting ω appropriately.
Eq.(3.31) cannot be solved analytically for the relationship between e and i for
frozen orbit solutions in this case and must be solved numerically. One possible
solution method is to use the θ integral and eliminate either e or i from Eq.(3.31).
However, since the range of θ is not immediately apparent, it is easier to choose
values of either e or i and solve Eq.(3.31) for the other orbital element.
Figure 3.4 shows the frozen orbit solutions for J2 6= 0 and J3 > 0. It is
immediately apparent that including the effect of J3 significantly increases the
complexity of the structure of the frozen orbit solutions. First of all, it is not
surprising that the structure of the frozen orbit solutions changes depending on the
sign of J3 sin ω. However, the symmetry returns for large values of the eccentricity
where the solutions are the same as in the tide-only case. This occurs since as the
43
eccentricity increases, the apoapsis of the orbit also increases and the influences of J2
and J3 on the dynamics decrease. Therefore, we conclude that the high-eccentricity
frozen orbits are not significantly changed by adding J2 and J3.
One of the new features apparent in Figure 3.4 is near-equatorial frozen orbits
that exist for all values of eccentricity for J3 sin ω > 0. We also observe that
circular frozen orbits exist only at four isolated points, which in this case have
inclinations of 0o, 180o, 63.43o and 116.56o. In contrast, in the tide-only and tide
plus J2 cases, circular frozen orbits existed over all inclinations. The exact values
of the non-equatorial inclinations for the four circular frozen orbits depend on the
parameters of the system. Those values will always be close to but not exactly equal
to i = sin−1(√
4/5). Their deviation from that value is due to the existence of the
tidal term. In the case where only J2 and J3 are included, these circular frozen
orbits exist independently of the parameter values [37]. Including the tidal term as
we do here breaks this relationship and the frozen orbits with i = sin−1(√
4/5) have
non-zero eccentricities.
The frozen orbits with eccentricities smaller than about 0.2 have very different
characteristics depending on whether J3 sin ω > 0 or J3 sin ω < 0. In terms of
Figure 3.4 where J3 > 0, frozen orbits with ω = π/2 have inclinations in the middle
range, between about 45o − 65o and 105o − 135o. Conversely, for frozen orbits with
ω = −π/2, near-circular frozen orbits exist for inclinations between about 65o− 105o
(i.e. near-polar). In addition, for eccentricities up to about 0.1 there exist frozen
orbits with inclinations ranging from circular to about 40o for direct orbits and in
the 140o − 180o range for retrograde orbits.
3.2.4 Variations in J3
The results discussed above were generated for specific values of J2 and J3. Since an
accurate value for J3 for Europa is not currently known, it is important to verify that
44
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
(a) Order of magnitude larger J3
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
(b) Order of magnitude smaller J3
Figure 3.5: Frozen orbit solutions with larger and smaller values of J3 (compared to Table 3.1).
the qualitative features of the results stay the same for different values of J3. This
would demonstrate that these results are applicable not only to Europa, whatever
its gravity coefficients are, but also to other planetary satellites. Figure 3.5 shows
the frozen orbits solutions using values of J3 an order of magnitude larger and an
order of magnitude smaller than the value given in Table 3.1. Although the shape of
the curves is slightly different from the curves in Figure 3.4, in both cases the same
classes of frozen orbits still exist. Therefore, even though the exact value of J3 for
Europa is not known at this time, the analysis of the system shown thus far should
be valid no matter what the value is and these results can be used in subsequent
analysis.
3.3 Stability of Frozen Orbit Solutions
To determine the stability of the frozen orbit solutions, linearize Eqs.(3.23)-(3.25)
about the frozen orbit solutions. First, express Eqs.(3.23)-(3.25) as:
de
dt= fe(e, i, ω) , (3.32)
di
dt= fi(e, i, ω) , (3.33)
45
dω
dt= fω(e, i, ω) . (3.34)
Define i = i∗ + δi, e = e∗ + δe and ω = ω∗ + δω where e∗, i∗ and ω∗ are the frozen
orbit values of these orbital elements. Then,
d(e∗ + δe)
dt= fe(e
∗ + δe, i∗ + δi, ω∗ + δω) , (3.35)
d(i∗ + δi)
dt= fi(e
∗ + δe, i∗ + δi, ω∗ + δω) , (3.36)
d(ω∗ + δω)
dt= fω(e∗ + δe, i∗ + δi, ω∗ + δω) . (3.37)
Expanding up to first order, we obtain:
e∗ + δe = fe(e∗, i∗, ω∗) +
∂fe
∂e
∣∣∣∣∗δe +
∂fe
∂i
∣∣∣∣∗δi +
∂fe
∂ω
∣∣∣∣∗δω , (3.38)
i∗ + δi = fi(e∗, i∗, ω∗) +
∂fi
∂e
∣∣∣∣∗δe +
∂fi
∂i
∣∣∣∣∗δi +
∂fi
∂ω
∣∣∣∣∗δω , (3.39)
ω∗ + δω = fω(e∗, i∗, ω∗) +∂fω
∂e
∣∣∣∣∗δe +
∂fω
∂i
∣∣∣∣∗δi +
∂fω
∂ω
∣∣∣∣∗δω . (3.40)
First, since the frozen orbit solutions are equilibrium solutions to Eqs.(3.32)-(3.34)
e∗ = fe(e∗, i∗, ω∗) = 0, i∗ = fi(e
∗, i∗, ω∗) = 0 and ω∗ = fω(e∗, i∗, ω∗) = 0. In addition,
since ω∗ = ±π/2:
∂fe
∂e
∣∣∣∣∗
=∂fe
∂i
∣∣∣∣∗
=∂fi
∂e
∣∣∣∣∗
=∂fi
∂i
∣∣∣∣∗
=∂fω
∂ω
∣∣∣∣∗
= 0 .
Therefore, Eqs.(3.38)-(3.40) simplify to the following:
δe =∂fe
∂ω
∣∣∣∣∗δω , (3.41)
δi =∂fi
∂ω
∣∣∣∣∗δω , (3.42)
δω =∂fω
∂e
∣∣∣∣∗δe +
∂fω
∂i
∣∣∣∣∗δi . (3.43)
46
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
Figure 3.6: Stability of frozen orbits about Europa for the tide-only case.
Next, taking the second derivative of Eq.(3.43) yields:
δω =
(∂fω
∂e
∣∣∣∣∗
∂fe
∂ω
∣∣∣∣∗+
∂fω
∂i
∣∣∣∣∗
∂fi
∂ω
∣∣∣∣∗
)δω . (3.44)
For each frozen orbit the partial derivatives evaluated at that frozen orbit are
constants. Therefore, δω is just a constant multiplied by δω. This second order
linear differential equation has the simple form:
δω = Λ2(e∗, i∗, ω∗)δω , (3.45)
where Λ2 is a constant for each frozen orbit.
The evaluation of Λ2(e∗, i∗, ω∗) for a particular frozen orbit determines its
stability. If Λ2 > 0 the frozen orbit is unstable and if Λ2 < 0 the frozen orbit is
stable. Figure 3.6 shows the stability of the frozen orbits for the tide-only case. Note
that all of the elliptic frozen orbits are stable and the circular frozen orbits are stable
for inclinations less than about 39.23o and greater than about 140.77o. The circular
frozen orbits with inclinations between ∼ 39.27o − 140.77o are unstable.
Figure 3.7 shows the stability of the frozen orbits for the tide plus J2 case. The
similarities between this and the tide-only case are that the elliptic frozen orbits
47
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 10
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
Figure 3.7: Stability of the frozen orbits for the tide plus J2 case.
with eccentricities larger than about 0.2 are stable. As well, the near-equatorial
circular frozen orbits are also stable. The major difference between this case and
the tide-only case is that the elliptic frozen orbits with eccentricities smaller than
about 0.2 are unstable as are the circular frozen orbits with inclinations ranging
from about 47o − 133o. The circular frozen orbits change from stable to unstable at
the bifurcation points (i.e., the locations where the elliptic frozen orbits bifurcate
from the circular frozen orbits).
The stability of the frozen orbits for the case including the tidal term, J2 and
J3 is similar to the tide plus J2 case in that the higher eccentricity frozen orbits are
stable. However, for the lower-eccentricity frozen orbits, the stability characteristics
of this case depend on the sign of J3 sin ω. For J3 sin ω > 0, the frozen orbits with
eccentricities less than about 0.2 are unstable. These are the mid-inclination frozen
orbits. In addition, the near-equatorial frozen orbits that exist for all inclinations are
also stable. On the other hand, for J3 sin ω < 0, the low-eccentricity, near-equatorial
frozen orbits are stable while the near-circular, near-polar frozen orbits are unstable.
The strength of the instability of an unstable frozen orbit can be measured by the
characteristic time. The characteristic time is defined as τ = 1/Λ, and is the time
48
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
Figure 3.8: Stability for frozen orbits about Europa that include the tidal term, J2 and J3.
interval it takes the eccentricity to increase by a factor of exp (1) ∼ 2.718[37]. In
general, the eccentricity will increase by an order of magnitude after exp (1) ∼ 2.718
characteristic times. Even though the tide-only frozen orbit solutions themselves do
not depend on the system parameters, the characteristic times of the unstable frozen
orbits are parameter-dependent. Consider Figure 3.9 which shows the characteristic
times for the 100 km altitude unstable circular frozen orbits for the tide-only cases of
Europa, Dione and Enceladus. Observe that the characteristic times for Europa are
the largest, followed by Dione and then Enceladus. This means that the frozen orbits
about Enceladus are the most unstable. In particular, the Enceladus characteristic
times are about an order of magnitude smaller than the Europa case. This has
strong implications for orbit design since a smaller characteristic time implies a more
highly unstable system in the sense that it will grow along its unstable manifold
more rapidly. All three of these planetary satellites are highly perturbed systems
in which science orbit design is very difficult, and the goals for the orbit design
will have to take the characteristic times into consideration. Observe in Figure 3.9
that the more polar the frozen orbit, the more unstable it is since it has a smaller
characteristic time.
49
40 50 60 70 80 90 100 110 120 130 1400
10
20
30
40
50
60
70
80
90
inclination (degrees)
time
(day
s)
(a) Europa
40 50 60 70 80 90 100 110 120 130 1400
5
10
15
20
25
30
inclination (degrees)
time
(day
s)
(b) Dione
40 50 60 70 80 90 100 110 120 130 1400
1
2
3
4
5
6
7
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(c) Enceladus
Figure 3.9: Characteristic times for the tide-only unstable frozen orbits with 100 km altitudes.
50
60 70 80 90 100 110 12013
13.5
14
14.5
15
15.5
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(a) Circular, near-polar
40 50 60 70 80 90 100 110 120 130 14010
20
30
40
50
60
70
80
90
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(b) Elliptic mid-inclination
Figure 3.10: Characteristic times for unstable frozen orbits about Europa
For the case including the tidal term, J2 and J3, there are two classes of unstable
frozen orbits, namely the near circular near polar frozen orbits with J3 sin ω < 0
and the mid-inclination frozen orbits with J3 sin ω > 0. Figure 3.10 shows the
characteristic times as functions of the inclination for both of these classes of
unstable frozen orbits about Europa. The characteristic times for the near-circular,
near-polar frozen orbits have the same structure as the circular frozen orbits for
the tide-only case, with the minimum characteristic time corresponding to a polar
orbit. The same trend is also apparent for the mid-inclination frozen orbits where
the closest to polar inclination also corresponds to the minimum characteristic time.
Just as we verified the frozen orbit solutions for different possible values of J3,
we also look at the stability and characteristic times for values of J3 one order of
magnitude smaller and one order of magnitude larger than the value in Table 3.1.
Figure 3.11 shows the stability and characteristic times for frozen orbits computed
with a J3 value one order of magnitude smaller and Figure 3.12 shows the same
plots computed with a J3 value one order of magnitude larger. We see that when
compared the the results for our nominal J3 case, the stability properties follow the
same trends. For the larger J3 value, the characteristic times are smaller, meaning
51
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
(a) Stability
65 70 75 80 85 90 95 100 105 110 11513.5
14
14.5
15
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(b) Characteristic times for circular, near-polar orbits
40 50 60 70 80 90 100 110 120 130 14010
20
30
40
50
60
70
80
90
100
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(c) Characteristic times for elliptic, mid incli-nation orbits
Figure 3.11: Stability and characteristic times of frozen orbits about Europa with J3 one order ofmagnitude smaller than Table 3.1
that the orbits are more unstable. This is expected since the J3 perturbation is
acting more strongly. For the smaller J3 value, the characteristic times are larger
than for the nominal J3 value case meaning that the orbits are more stable.
52
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
0
20
40
60
80
100
120
140
160
180
esinω
incl
inat
ion
(deg
rees
)
unstablestable
(a) Stability
65 70 75 80 85 90 95 100 105 110 1157
8
9
10
11
12
13
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(b) Characteristic times for circular, near-polar orbits
40 50 60 70 80 90 100 110 120 130 14010
12
14
16
18
20
22
24
26
28
inclination (degrees)
char
acte
ristic
tim
e (d
ays)
(c) Characteristic times for elliptic, mid-inclination orbits
Figure 3.12: Stability and characteristic times of frozen orbits about Europa with J3 one order ofmagnitude larger than Table 3.1
.
53
3.4 Contour Plots and Frozen Orbit Integrations
In this section, the secular motion in the vicinity of frozen orbits will be examined
using contour plots. Then, results of integrations in the 3-DOF system of frozen
orbit solutions will be compared with the contour plots to determine if the stable
frozen orbits are viable orbits in the full 3-DOF system and how unstable frozen
orbits behave in the 3-DOF system. All of the results in this section are computed
for the Europa system, with the tidal component, J2, C22 and J3 gravity field terms,
but are applicable to other planetary satellite systems.
3.4.1 Contour Plots
Contour plots are a tool that can be used to visualize the secular motion of an
orbiter. They are plotted in the 1-DOF system which is integrable and a function
of only two variables, (G,ω) and so a plot in terms of those two variables can
completely describe the motion. It is useful, however, to produce contour plots for
other sets of variables in order to get a better picture of the motion. Recall that
G =√
µa(1− e2). Therefore, a plot in terms of (e,ω) can also be used to describe
the motion. The final contour plot that we use is in terms of the semi-equinoctial
variables (h,k). Although we initially compute these plots for a range of frozen
orbits, they are most useful to describe near-circular motion, as will be shown later.
Note that contour plots of the set (i,ω) are also possible. However, since the motion
is integrable, the inclination can be determined from the integral of motion θ and
the eccentricity. We are, in general, more interested in the motion of the eccentricity
than the inclination since both remain relatively constant for stable orbits, and
for unstable orbits it is the growing eccentricity that causes an impact with the
planetary satellite.
The contour plots are made up of curves with constant values of the 1-DOF
potential, R. The curves define the path that the motion will follow on average.
54
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
−3000 −2000 −1000 0 1000 2000 3000
−3000
−2000
−1000
0
1000
2000
3000
Gcosω
Gsi
nω
Figure 3.13: Contour plots for motion in the vicinity of a stable frozen orbit with e∗ = 0.548,i∗ = 50o, J3 sinω > 0.
55
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
Figure 3.14: Contour plots for motion in the vicinity of an unstable frozen orbit with e∗ = 0.0.039,i∗ = 55o, J3 sinω > 0.
56
Figures 3.13-3.16 show contour plots in all three regimes for motion in the vicinity
of four different types of frozen orbits. Figure 3.13 is an example of motion in the
vicinity of a stable frozen orbit with an inclination of 50o, an eccentricity of about
0.548 and J3 sin ω > 0. It is, therefore, a mid-inclination, mid-eccentricity frozen
orbit. We see that for this case the motion is symmetric for ω = ±π/2, showing
that the tidal effect dominates over the J2 and J3 effects. Figure 3.14 shows motion
in the vicinity of an unstable frozen orbit with an inclination of 55o, an eccentricity
of about 0.039 and J3 sin ω > 0. In this case, the motion is not symmetric about
ω = ±π/2 since the eccentricity is small and the J2 and J3 effects are important.
Note that there appears to be two stable regions in the plot, other than the unstable
frozen orbits. These regions correspond to equilibrium points in the 1-DOF system
that do not have the altitude of periapsis of 100 km that we consider. Their locations
correspond to motion below the surface of the planetary satellite since the contour
plots have a constant value of the semi-major axis. For this case, the semi-major
axis is a = rp/(1 − e) = 1660.8/(1 − 0.039) = 1728 km and so motion below the
surface of the planetary satellite occurs for eccentricities greater than 0.097. The
consequence of this result is that the only motion that we are interested in from a
contour plot is that in the vicinity of the frozen orbit.
Now, consider contour plots for frozen orbits with J3 sin ω < 0. First, Figure 3.15
shows motion in the vicinity of a stable frozen orbit with an inclination of 30o and
an eccentricity of about 0.047. Again, the motion is not symmetric about ω = ±π/2
since this is a low-altitude orbit where the effects of J2 and J3 are significant. The
last example, in Figure 3.16, is an unstable frozen orbit with an inclination of 80o
and an eccentricity of 0.021. Note that the plot for (e,ω) is not over the entire range
of eccentricity but just in the region of the frozen orbit. This gives a better picture
of the motion in the vicinity of the frozen orbit and as previously noted, motion
for larger values of the eccentricity takes place under the surface of the planetary
57
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
k=ecosω
h=es
inω
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
Figure 3.15: Contour plots for motion in the vicinity of a stable frozen orbit with e∗ = 0.047,i∗ = 30o, J3 sinω < 0.
58
0 50 100 150 200 250 300 3500
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
ω (degrees)
ecce
ntric
ity
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
−2000 −1500 −1000 −500 0 500 1000 1500 2000
−2000
−1500
−1000
−500
0
500
1000
1500
2000
Gcosω
Gsi
nω
Figure 3.16: Contour plots for motion in the vicinity of an unstable frozen orbit with e∗ = 0.021,i∗ = 80o, J3 sinω < 0.
59
satellite. The frozen orbit in this example is near-circular and near-polar, and it is
the type that will be considered later as a possible science orbit for a mission to a
planetary satellite.
3.4.2 Frozen Orbit Integrations
Results of integrations of frozen orbit solutions are presented overlaying contour
plots in (e,ω) space for both stable and unstable frozen orbits. For stable frozen
orbits, the goal is to determine whether they are viable long-term stable orbits in
the 3-DOF system. For unstable frozen orbits, we are interested in comparing the
motion in the 3-DOF system with the contour plots to see if the motion follows the
contours well.
Figure 3.17(a) shows a contour plot in (e,ω) space along with the results of the
integration in the 3-DOF system of a stable frozen orbit with an inclination of 134o,
an eccentricity of 0.387 and J3 sin ω > 0. The integration takes place over a period
of 150 days and it is clear that the orbit is stable since it stays in the libration
region. Note that the center of the circle representing the integrated trajectory is
not exactly at the frozen orbit location. This is due to the perturbations that are
averaged out in the 1-DOF system. Even though the integrated trajectory does not
follow the curves in the contour plot exactly, the frozen orbit conditions still yield a
stable orbit in the 3-DOF system.
The next example, shown in Figure 3.17(b), shows the integration of a stable
frozen orbit with an inclination of 62o, an eccentricity of 0.795 and J3 sin ω > 0.
The integrated trajectory, shown by the red curve, does not follow the curves in the
contour plot at all and reaches a very high eccentricity which leads to an impact with
the planetary satellite. This occurs since although the orbit is stable in the 1-DOF
system, it has a large eccentricity which means that it has a large apoapsis radius.
Since the motion takes place farther away from the planetary satellite, perturbations
60
not taken into account in the derivation of the 1-DOF arise and cause the 3-DOF
trajectory to be unstable.
Finally, Figure 3.17(c) is the integration of an unstable frozen orbit with an
inclination of 80o, an eccentricity of 0.021 and J3 sin ω < 0. Since this orbit is
unstable, all that we expect is the trajectory to follow the contour curve starting at
the frozen orbit location. This is observed, allowing us to conclude that for small
eccentricities the contour plots give a very good estimate of what motion to expect in
the 3-DOF system. Therefore, the motion in the 1-DOF system can be used to plan
orbits for a mission to a planetary satellite, as will be seen in subsequent chapters.
61
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
(a) e∗ = 0.387, i∗ = 134o, J3 sinω > 0
0 50 100 150 200 250 300 3500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
ecce
ntric
ity
(b) e∗ = 0.795, i∗ = 62o, J3 sinω > 0
0 50 100 150 200 250 300 3500
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
ω (degrees)
ecce
ntric
ity
(c) e∗ = 0.021, i∗ = 80o, J3 sinω < 0
Figure 3.17: Contour plots and integrations in the 3-DOF of various frozen orbits
62
CHAPTER 4
ROBUST CAPTURE AND TRANSFER
TRAJECTORIES
In this chapter, we study capture trajectories in the Hill 3-body problem.
Capture trajectories begin in the exterior region of Hill’s problem, enter the interior
region and orbit the planetary satellite at least once. One particular feature of
capture trajectories that we investigate is their lifetime. Uncontrolled, these orbits
can impact with the planetary satellite or exit the interior Hill region after very
short time spans. Koon et al. [20] investigated orbits that travel between the interior
and exterior Hill regions and showed that the amount of time a trajectory spends
orbiting the planetary satellite is determined by chaotic dynamics.
A method is developed which identifies sets of capture trajectories that do
not impact or escape the planetary satellite for extended time periods. These are
called ‘safe trajectories’ and the regions in which they lie ‘safe zones’. These safe
low-energy trajectories may be useful for a future mission to a planetary satellite
such as Europa. With the science goals of that type of mission being such that a
low-altitude, high inclination stable orbit about Europa is desirable, we examine
low-cost methods to transfer from a safe capture trajectory to a long-term stable
orbit such as the elliptic frozen orbits developed in the last chapter.
Criteria on the safe capture trajectories that result in the lowest cost transfers
are determined and we develop schemes to transfer to elliptic frozen orbits and
circular frozen orbits. These schemes are analyzed based on their costs and specific
63
examples are shown. The end result is a method that identifies trajectories that
enter into orbit about Europa on low-energy capture trajectories and which do not
impact or escape for at least one week, and from which it is possible to transfer to a
more stable orbit with relatively low cost.
4.1 Non-dimensional Hill 3-Body Problem and
Libration Points
In this chapter, only the tidal term is included in the perturbing potential R, given
in Eq.(3.1), yielding the following potential:
R =1
2N2(3x2 − r2
). (4.1)
This is an acceptable simplification since we are primarily dealing with motion
farther away from Europa than the low-altitude frozen orbits in which the gravity
coefficients are important. Although the capture trajectories to be presented in this
chapter are in the vicinity of Europa, if they do pass very closely to Europa, it is
only for a short period of time. In addition, when we consider transfers from capture
trajectories to frozen orbits, we consider either the elliptic frozen orbits or higher
altitude circular orbits, both of which are not affected very strongly by J2, C22 or J3.
The equations of motion for this system are:
x = 2Ny − µ
r3x + 3N2x , (4.2)
y = −2Nx− µ
r3y , (4.3)
z = − µ
r3z −N2z , (4.4)
and they are in dimensional form. However, to simplify the computations in this
chapter we make the model non-dimensional. To do this, take l = (µ/N2)1/3
as the
unit length and τ = 1/N as the unit time. The equations then take on the following
64
−0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x
y
Forbidden Region
Forbidden Region
L2 L
1
Figure 4.1: Regions of allowable motion and libration points for J = −2.15. The dotted linerepresents a planar capture trajectory that originates near L2.
form
x = 2y − x
r3+ 3x, , (4.5)
y = −2x− y
r3, (4.6)
z = − z
r3− z , (4.7)
and have no free parameters.
Equilibrium solutions exist in the Hill 3-body problem, analogous to the L1 and
L2 libration points in the CR3BP. There are two libration points that are symmetric
about the origin with coordinates x = ± (µ/3N2)1/3
, y = z = x = y = z = 0
where the x-coordinate in nondimensional form is x = ± (1/3)1/3 = ±0.693 · · · (see
Figure 4.1).
Recall that Eqs. (4.5)-(4.7) have the Jacobi integral, which in non-dimensional
form for the tide-only case is:
J =1
2v2 − 1
r− 1
2(3x2 − z2) , (4.8)
where v =√
x2 + y2 + z2 is the speed of the spacecraft in the rotating frame. The
condition v ≥ 0 in Eq.(4.8) places a restriction on the position of the particle for
a given value of J . Setting v = 0 defines the zero-velocity surface, which sets a
65
physical boundary for the motion (see Figure 4.1).
For values of the Jacobi constant above a critical value, it is possible for both
escaping and capture trajectories to exist in this problem. This critical value of J
defines the energy at which the zero-velocity surfaces open at L1 and L2 and is equal
to
Jcrit = −1
292/3 = −2.16337 . (4.9)
Since our goal is to characterize capture trajectories, we restrict ourselves to
trajectories with a Jacobi constant greater than Jcrit.
4.2 Periapsis Poincare Maps
A Poincare map associates a continuous time dynamical system to a discrete time
system. The use of a Poincare map reduces the dimensionality of a system by at
least one, and by two if there exists a first integral in the system, as is the case in the
Hill 3-body problem. In general, Poincare maps are defined such that the surface of
section is a plane in position space, such as x = 0. However, since all that is needed
to define a Poincare map are two surfaces of section that are transversal to the flow,
they are defined differently here and periapsis passages are used as the surface of
section, following [42].
4.2.1 Definition of the Map
Our Poincare map relates capture trajectories that start near the libration point
L2 to the periapsis passages of these trajectories. Following [41], the initial surface
of section is defined as the surface of a sphere with radius (13)1/3 bounded by the
zero-velocity surface. This surface passes through the libration point region and
extends as far as the zero-velocity surface of the Hill problem. The image surface is
defined by the periapsis condition r = 0 and r > 0. For the 3-D Hill problem, the
phase space is six-dimensional. The use of the Poincare map restricts the problem
66
to the surface of section, and hence reduces the problem by one dimension. The
Poincare map is then computed at a given value of the Jacobi integral J , (and so
the map is a function of J), reducing the problem by one further dimension to four
dimensions.
Due to the symmetry between the L1 and L2 libration points, analysis is only
performed on capture trajectories that originate near L2; however, the results
can be directly related to trajectories that originate near L1 by the following
transformation[42]
(x, y, z, x, y, z, t)Gµ−→ (−x,−y,−z,−x,−y,−z, t) (4.10)
since the equations of motion remain unchanged. By a capture trajectory, we mean a
trajectory that remains in the Hill region of the planetary satellite for a finite period
of time. These trajectories enter the region of the planetary satellite in the vicinity
of the libration point and orbit the planetary satellite at least once. Figure 4.1 shows
an example of a capture trajectory.
In the 3-D case, the periapsis Poincare map is four dimensional and four
parameters are needed to characterize it. For the initial surface of section, we use
the (x, z) coordinates and two angles (δ, φ) for the direction of the velocity vector.
The (x, z) coordinates are chosen randomly on the section of the surface of the
sphere that falls within the allowable region. Since the surface of section is a sphere,
the y coordinate can be computed from the relationship r =√
x2 + y2 + z2, where
r = (1/3)1/3 and the sign for y is chosen randomly. The initial conditions are then:
x0 = x x0 = v cos φ cos δ
y0 = ±√
r2 − x2 − z2 y0 = v cos φ sin δ (4.11)
z0 = z z0 = v sin φ
67
where (φ, δ) ∈ [π/2, 3π/2], and v is computed from the Jacobi integral:
v =
√2
(J +
1
r
)+ (3x2 − z2) . (4.12)
The parameters used to represent the image map are the periapsis position vector
(x, y, z) and the inclination.
With the initial conditions given by Eq.(4.11), an 8(7) order Runge-Kutta-
Fehlberg integration routine is used to integrate the trajectories. They are integrated
for multiple passages through the image map, meaning that they have multiple
periapsis passages. The phase space is studied in terms of these multiple periapsis
passages for various values of J .
4.2.2 Poincare Map Results
As an initial study, consider the planar case of the Hill 3-body problem. In this
model, motion is restricted to lie in the x-y plane, and the periapsis Poincare
map reduces to two dimensions. The initial surface, which consists of the surface
of a sphere in the three-dimensional case reduces here to an arc of circle. It is
parameterized with the x coordinate and an angle δ which defines the direction of
the velocity vector. This is equivalent to the initial conditions defined by Eq.(4.11)
with z = φ = 0. On the image map, the coordinates (x, y) of the periapsis position
vector are used.
Although the periapsis Poincare map is computed in non-dimensional coordinates
to allow for its application to many different physical systems, we primarily consider
Europa. On many of the periapsis Poincare maps, the surface of Europa is indicated
to give perspective to the plot. Although some of the periapsis passages of these
trajectories lie beneath the surface of Europa, indicating that they impact with it, for
the time being they are included in the analysis since they are not necessarily impact
trajectories when a different planetary satellite is considered. In a later section of
68
(a) First periapsis passage region
(b) First 4 periapsis passage regions: Black - 1st, blue- 2nd, red - 3rd, green - 4th
Figure 4.2: Periapsis Poincare maps for J=-2.15 (planar case) where the circle represents thesurface of Europa.
this chapter, we distinguish between impacting and non-impacting trajectories.
We first consider the periapsis Poincare map for the first periapsis passage of
trajectories with J = −2.15, as was done in [42]. As shown in Figure 4.2(a) the first
periapsis passages are divided into two disjoint regions. The first, located near the
planetary satellite, corresponds to trajectories that immediately enter the region in
the vicinity of the planetary satellite and have their first periapsis passage there.
The second region, located near the L2 libration point, corresponds to trajectories
that do not immediately leave the vicinity of L2 and have their first periapsis passage
in that region. These can be associated with the periodic orbit that exists about L2,
as shown in [41]. Some of these trajectories will subsequently enter the region of the
planetary satellite and some will escape from the system entirely. Note that by our
current definition of capture trajectory, there is no guarantee that the trajectory
actually comes from outside the Hill region and not from inside the region itself.
To remedy this, an additional integration is performed whereby the trajectory is
integrated backwards in time for four time units. This is a sufficient amount of
69
time for a true capture trajectory to return to its starting location outside of the
Hill region. Therefore, the set of capture trajectories is restricted to those that lie
beyond the boundary of the Hill region after a backwards-time integration of four
time units. If after the backwards-time integration a trajectory was located inside
the Hill region, we could conclude that it is not a true capture trajectory, but just
a trajectory that happened to have an inward velocity at the opening of the Hill
region at the initial time.
We now extend the periapsis Poincare map to four periapsis passages by plotting
the first four periapsis passages of each trajectory for J = −2.15. Each periapsis
passage is assigned a different color, as shown in Figure 4.2(b). Observe that
each subsequent periapsis passage region becomes more spread out. This occurs
since symplectic coordinates are not used on our map. In addition, some of the
colors overlap (i.e, there is a small blue region inside the black region). These
overlaps occur because, as previously mentioned, some of the trajectories do not
immediately have a periapsis passage in the vicinity of the planetary satellite but
instead have their first periapsis passage near the L2 region. Therefore it is not until
the second periapsis passage that these trajectories are near the planetary satellite.
The dynamics of these trajectories, once they enter the region near the planetary
satellite, are the same as the dynamics of the trajectories that enter that region
immediately. It is also important to note that the gaps in some of the regions are not
because trajectories cannot have periapsis passages there, but because these regions
were generated numerically and, as such, only a finite number of trajectories can be
considered. From this point forward, the region of first periapsis passages that occur
in the vicinity of the L2 libration point will be ignored. The first periapsis passage
of a particular trajectory is defined to be its first periapsis passage that occurs in
the vicinity of the planetary satellite.
To illustrate how the periapsis Poincare maps depend on J , Figure 4.3 shows
70
Figure 4.3: Poincare maps for various values of J . The circle in each plot denotes the surface ofEuropa.
71
−0.12−0.11
−0.1−0.09
−0.08−0.07
−0.06−0.05
0
0.01
0.02
0.03
0.04
0.05−0.015
−0.01
−0.005
0
0.005
0.01
0.015
xy
Figure 4.4: First periapsis passage region for J=-2.15 (3-d case). The position is indicated by thebase of the arrow, the velocity direction by the direction of the arrow, and the magnitude of thevelocity by the length of the arrow.
a series of Poincare maps for increasing values of J . The regions become larger
as J increases, which is expected since the measure of the initial condition region
increases with J . Only the first periapsis passage regions are shown, since the trends
shown by these regions also apply to subsequent periapsis passages. Note that the
plots in Figure 4.3 are mirror images of the corresponding plots shown in [42].
It is much more difficult to visualize the periapsis Poincare maps for the 3-D
problem since the Poincare map itself is four dimensional. In general, the trends we
see in the planar case apply to the non-planar case, such as the periapsis passage
regions becoming larger in position space as J is increased as well as for subsequent
periapsis passages. One way to visualize the Poincare map for the non-planar case
is to plot each periapsis passage’s three-dimensional position vector, with an arrow
indicating the direction of its velocity (representing the inclination at the periapsis
passage). An example of this is shown in Figure 4.4. Another way to visualize
the non-planar case is by considering the inclination of the periapsis passage as
a function of its radius. Figure 4.5(a) shows the inclination as a function of the
radius for the first periapsis passage of trajectories with various values of J . Observe
that as J increases, the periapsis passages reach higher inclinations. Figure 4.5(b)
72
(a) First periapsis passage regions, various values ofJ
(b) First 4 periapsis passages, J = −2.15
Figure 4.5: Inclination as a function of normalized radius for 3-d capture trajectories.
73
shows the inclination as a function of the radius for multiple periapsis passages of
trajectories with J=-2.15. In this case, the maximum obtainable inclination does
not increase for subsequent periapsis passages, but the periapsis passages do occur
over larger ranges of radii.
We now introduce a more systematic description of these maps and regions.
Define the set of trajectories that lie in the first periapsis passage region close to
Europa as the set S1,
S1 = x|x ∈ Initial Region . (4.13)
Then, under the flow dynamics of the Hill problem the Poincare map can be
represented as Φ, where
Sn+1 = Φ(Sn) = Φ2(Sn−1) = · · · = Φn(S1) . (4.14)
Thus, it is clear that any point in Sn+1 has a unique image in S1. If we use the
symmetry operator defined by Eq.(4.10), Gµ, we can also associate the ‘mirror image’
trajectories as S ′n = Gµ(Sn) defined by reflecting all coordinates and velocities about
the origin.
4.3 Symmetry Between Escape and Capture
Trajectories
Aside from the symmetry Gµ between L1 and L2 which was previously discussed,
additional symmetries exist in this problem (and in the CR3BP). Two that are
considered relate to the symmetry between escape and capture trajectories. If
(x, y, z, x, y, z, t) is a solution of the equations of motion, then the trajectories
obtained by applying the following transformations are also solutions [42]:
(x, y, z, x, y, z, t)G1−→ (−x, y, z, x,−y,−z,−t) , (4.15)
(x, y, z, x, y, z, t)G2−→ (x,−y, z,−x, y,−z,−t) . (4.16)
74
(a) First periapsis passage regions for capture trajec-tories and symmetric escape regions
(b) First 4 periapsis passage regions with symmetricescape regions, black - 1st, blue - 2nd, red - 3rd, green- 4th
Figure 4.6: Poincare maps showing symmetric capture and escape regions for J = −2.15.
Thus, applying either of the two above symmetries to a capture trajectory will
yield an escaping trajectory. An escaping trajectory is a trajectory that crosses the
initial surface of sphere with an outward velocity and exits the Hill region. Figure
4.6 shows the symmetry between the capture and escape regions on the Poincare
periapsis map for the first periapsis passage in the planar case. This figure shows the
first periapsis passage region for the capture trajectories, and hence for the escaping
trajectories this corresponds to the periapsis passage immediately preceding escape
(or transfer to the region in the vicinity of L1,2).
This symmetry between capture and escape trajectories can be extended past
75
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
x
y L2 L
1
Forbidden Region
Forbidden Region
Figure 4.7: A capture trajectory that escapes after 4 periapsis passages. The periapsis passages areindicated by ‘x’.
the first periapsis passage. This allows us to determine when capture trajectories
can escape from the Hill region. A capture trajectory escapes the Hill region when
it passes through the initial surface of sphere with an outward velocity. However,
a captured trajectory can not escape arbitrarily. We found that it will only escape
when it passes through the ‘first periapsis passage’ escape region, which is the
escaping region symmetric to the first periapsis passage capture region as shown on
Figure 4.6(a). For this example (J=-2.15), we found that the earliest that capture
trajectories pass through this region is during their third periapsis passage, and so
we conclude that once a capture trajectory with J=-2.15 is found, it is guaranteed
to have at least 3 periapsis passages prior to escape. Such a trajectory must also
lie in the pre-image of the escape region. Due to this, these transfer regions can
be easily found by identifying regions where these symmetric regions intersect with
each other. Figure 4.6(b) is a periapsis Poincare map which shows the first four
periapsis passages of capture trajectories as well as one set of their symmetric
regions corresponding to escaping trajectories. We see that there is an overlap
of these regions, demonstrating the mechanism by which capture trajectories can
subsequently escape. An example of this is shown in Figure 4.7. This trajectory has
its third periapsis passage in the portion of the red region that overlaps with the blue
symmetric region. Therefore, its fourth periapsis passage will be in the portion of the
76
green region that overlaps with the black symmetric region, as seen in Figure 4.6(b).
Any capture trajectory which has a periapsis passage in the black symmetric region
will not have any further periapsis passages and will escape. Thus, from the fact
that this trajectory had its third periapsis passage in a region that overlaps with the
blue symmetric escaping region, it will have a total of four periapsis passages and
then escape. As seen in Figure 4.7, this trajectory does indeed escape from the Hill
region after the fourth periapsis passage.
The symbolic notation defined earlier is used to give a clear description of this
result. First, let the time reversal symmetry be the operator Gt. From the property
Figure 4.11: Periapsis passages of capture trajectories for J = −1.70 with altitudes ≤ 250 km.
from the surface of Europa.
Recall that the elliptic frozen orbits for the tide-only case have orbit elements
ω = ±π/2, and e =√
1− 5/3 cos2 i. Also note that for the larger eccentricity frozen
orbits being considered here, the gravity field of the planetary satellite does not
strongly affect the frozen orbit solutions, justifying our use of the tide-only case.
Since the goal is to transfer from a safe capture trajectory to a frozen orbit, it is
necessary to determine what characteristics the periapsis passages of the capture
trajectories must have in order to perform this maneuver. The first step is to only
consider periapsis passages of capture trajectories that have an argument of periapsis
close to the frozen orbit value. Since the frozen orbits oscillate in the unaveraged
system, the condition ω = ±90o can be relaxed somewhat to consider periapsis
passages of safe capture trajectories that fall within 5 degrees of ω = ±90o. Then, if
the periapsis passage of the safe capture trajectory has an inclination for which an
elliptic frozen orbit exists, a maneuver can be performed to change the eccentricity
so it corresponds with the inclination based on the frozen orbit relation, placing the
spacecraft in an elliptic frozen orbit.
Figure 4.12 shows the set of periapsis passages of safe capture trajectories with
83
J=-1.60 that lie within 250 km of the surface of Europa and have an argument of
periapsis within 5 degrees of ±90o. A line denoting where the elliptic frozen orbits
lie is shown, which makes apparent the change in eccentricity required to transfer
from a capture trajectory to a frozen orbit. Note that the frozen orbit line actually
passes through the safe zone. This means that capture trajectories that lie on the
frozen orbit line have orbital elements corresponding to a frozen orbit at that instant.
However, a true elliptic frozen orbit is stable and bounded and would not arrive from
outside of the Hill sphere. This implies that the frozen orbit assumptions are not
valid at this point. Frozen orbits at higher eccentricities would be subject to larger
perturbations and so would not be valid either. We do not know precisely where the
averaging assumptions fail, however, we have evidence from numerical integrations
that points to the left of the frozen orbit line are still governed by the averaged
equations of motion. Therefore, when considering safe capture trajectories from
which to initiate a transfer, we do not consider periapsis passages that fall to the
right of the frozen orbit line, since this is beyond the point at which the averaging
assumptions break down.
Figure 4.12 also shows the cost, in meters per second, to transfer from the safe
capture trajectory to an elliptic frozen orbit. The periapsis passages of safe capture
trajectories used to initiate the transfer are those shown in Figure 4.12. The region
in the figure denoted ‘not valid’ refers to the region where the frozen orbits have
broken down. The region denoted ‘valid’ refers to the fact that these transfers are
practical since the frozen orbits exist. The gray region is present since we do not
have an absolute boundary on where the frozen orbits cease to exist, and so some
transfers in the gray region may be valid while others may not.
To prove that our scheme to transfer to an elliptic frozen orbit works, we
computed the transfer for one of the periapsis passages of the safe capture
trajectories and then continued the integration to show that the resulting trajectory
84
48 50 52 54 56 58 60 62 64
0.6
0.65
0.7
0.75
0.8
0.85
inclination (degrees)
ecce
ntric
ity
frozen orbits
∆v
(a) (b)
Figure 4.12: Potential periapsis passages with J = −1.60 for transfers to elliptic frozen orbits andcosts. (a) Periapsis passages with altitudes ≤ 250 km ω = ±90o ± 5o. (b) Cost to transfer to acorresponding elliptic frozen orbit.
is an elliptic frozen orbit. Starting from the safe capture trajectories with J = −1.60,
as shown in Figure 4.12, we choose the periapsis passage that has the following
parameters: e = 0.718, ω = −93.66o, i = 51.14o, and a periapsis altitude of 166 km.
The eccentricity that corresponds to a frozen orbit at this inclination is 0.586, and
so the cost of performing this eccentricity change is 69 m/s. To show that the new
orbit is an elliptic frozen orbit, we plot the trajectory on contour plots of eccentricity
as a function of argument of periapsis and inclination as a function of argument of
periapsis. Figure 4.13 shows that the new orbit does stay inside the libration region
of the contour plot,verifying the successful transfer from a safe capture trajectory to
a bounded orbit close to frozen orbit conditions.
4.5.3 Transfers to Circular Frozen Orbits
Two approaches to transferring to a circular frozen orbit are considered: two-impulse
and one-impulse maneuvers. The two-impulse scheme involves performing a
maneuver at a periapsis passage of the capture trajectory that allows us to use
the dynamics of elliptic frozen orbits to achieve a lower eccentricity (which in turn
increases the inclination). Following that, a second maneuver is performed at the
85
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
e
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
10
20
30
40
50
60
70
80
90
ω (degrees)
i (de
gree
s)
Figure 4.13: Contour plots and the numerical results for the orbit obtained after the transfermaneuver.
much lower eccentricity to circularize the orbit. The one-impulse maneuver involves
circularizing directly from the periapsis passage of the capture trajectory. Note that
this analysis assumes that the circular frozen orbits in the tide-only case exist. For
low-altitude orbits, we showed in Chapter 3 that the planetary satellite’s gravity field
has a large effect on the motion. When the effect of the planetary satellite’s gravity
field is included, the previously circular frozen orbits become near-circular, and have
small eccentricities. However, this does not impact our analysis very strongly since
we compute costs to circularize, which can be considered as a worst-case assumption
that is more costly than transferring to a near-circular orbit.
Consider first the two-impulse approach. Previously, in order to transfer to an
elliptic frozen orbit, a maneuver was chosen to achieve the frozen orbit eccentricity.
By examining the left plot in Figure 4.13 we see that if a slightly higher eccentricity
is chosen (i.e. an eccentricity near the top of the libration region), much larger
variations in eccentricity are achieved, to the point where the eccentricity occasionally
gets very close to 0. Figure 4.14 shows this phenomenon, where instead of performing
a maneuver to change the eccentricity to 0.586 as above, the eccentricity is changed
to 0.687. The same periapsis passage as in the elliptic frozen orbit transfer example
is used (i.e, e = 0.718, i = 51.14o, ω = −93.66o, altitude=166km). The cost of this
86
maneuver is 38.6 m/s. The next step in this approach is to circularize the orbit.
Depending on the altitude of the orbit after it is circularized, some control maneuvers
may be necessary to keep the orbit from drifting too much. The frequency and size
of these maneuvers will depend on how much of a drift we can tolerate and how
quickly the orbit is drifting (depending on its altitude).
The circularization maneuver is performed at a periapsis passage of the orbit
shown in Figure 4.14. In order to understand how the radius, eccentricity and
inclination of the orbit vary and where the periapsis passages occur, refer to
Figure 4.15 which shows the radius, eccentricity and inclination as functions of
time. The periapsis passages are indicated with a red ‘x’ and the solid blue line in
the top plot represents the surface of Europa. Observe that a lower eccentricity is
correlated to a higher radius and higher inclination. Higher inclinations are more
desirable; however, the higher radius associated with the higher inclination will lead
to an initially large circular orbit. To compare these issues, two examples will be
provided: one where the circularization maneuver is performed from the minimum
eccentricity of the orbit shown in Figure 4.14 and one where the circularization is
performed from a higher eccentricity. In both cases, the radius of the orbit will be
relatively large and some control maneuvers will be necessary to maintain this orbit.
Table 4.1 shows the cost of achieving both of these circular orbits as well as their
average radius and initial inclination. Figure 4.16 shows their characteristics.
We now consider the one-impulse approach for transferring to a circular orbit.
This approach involves directly circularizing the orbit at the periapsis passage of
the capture trajectory. The cost of this maneuver is 444.5 m/s and the resulting
orbit has a radius of 1731 km which corresponds to an altitude of 166 km and an
inclination of 51.14o.
All of the transfers to circular frozen orbits computed thus far result in orbits
that have different radii and inclinations. To compare them, it is necessary to
87
Table 4.1: Comparison of costs to transfer from a capture trajectory to a circular frozen orbit.
Example 1a Example 2b
From capture trajectory to near-frozen elliptic orbit 38.6 m/s 38.6 m/s
Circularization 10.3 m/s 230.3 m/s
Total Cost 48.9 m/s 268.9 m/s
Average Radius 5300 km 2875 km
Initial Inclination 61.6o 57.9o
a Circularizing at the minimum eccentricity of the near-frozen elliptic orbitb Circularizing at higher eccentricity
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω (degrees)
e
−180 −160 −140 −120 −100 −80 −60 −40 −20 00
10
20
30
40
50
60
70
80
90
ω (degrees)
i (de
gree
s)
Figure 4.14: Contour plots and numerical results for a transfer orbit to a circular frozen orbit.
88
0 5 10 15 20 250
0.2
0.4
0.6
0.8
e
0 5 10 15 20 2545
50
55
60
65
i (de
gree
s)
0 5 10 15 20 250
5000
10000
r (k
m)
time (days)
Figure 4.15: Characteristics of the near-frozen elliptic transfer orbit as functions of time. Eachperiapsis passage is represented by an ‘x’ and the solid line in the top plot represents the surface ofEuropa.
0 1 2 3 4 5 6 70
0.05
0.1
0.15
0.2
e
0 1 2 3 4 5 6 758
60
62
64
i (de
gree
s)
time (days)
0 1 2 3 4 5 6 74500
5000
5500
6000
r (k
m)
(a)
0 1 2 3 4 5 6 70
0.01
0.02
0.03
e
0 1 2 3 4 5 6 755
56
57
58
i (de
gree
s)
time (days)
0 1 2 3 4 5 6 72800
2850
2900
2950
r (k
m)
(b)
Figure 4.16: Characteristics of both possible circularized orbits over one week. (a) Transfer fromthe minimum eccentricity of the near-frozen elliptic orbit. (b) Transfer from a higher eccentricityof that same orbit.
89
Table 4.2: Comparison of costs to transfer from capture trajectory to tightly bound circularorbit
Method
Using Dynamics to Circularize Direct Circularization
Example 1 (i = 61.6o) 591.5 m/s 692.2 m/s
Example 2 (i = 57.9o) 586.6 m/s 604.7 m/s
a Circularization takes place at the minimum eccentricity of near-frozen elliptic orbitb Circularization takes place at a higher eccentricity
consider additional transfers so that a common orbit is achieved for both approaches.
Two common orbits are considered, with inclinations of 61.6o and 57.9o and a radius
of 1731 km. The two inclinations correspond to the inclinations achieved by the
two examples of the two-impulse approach and the radius corresponds to the radius
achieved by the one-impulse approach. The orbit obtained by the one-impulse
approach has an inclination of 51.14o and so to transfer to inclinations of 61.6o and
57.9o costs 247.7 km/s and 160.2 km/s, respectively. The radii of the orbits achieved
by the examples for the two-impulse approach are about 5300 km and 2875 km and
the costs of the Hohmann transfers to reduce those radii to 166 km are 542.6 m/s
and 299.7 m/s respectively. Table 4.2 shows a comparison of the costs of the two
approaches. We see that the method which uses the dynamics of the system in the
vicinity of the elliptic frozen orbits is more efficient, and the difference between the
two methods is larger when the final inclination is larger.
If the total costs of only the two-impulse approach are compared, we see that
although the second example is slightly cheaper, an inclination of almost 4o lower
is achieved. If the Hohmann transfer component of this approach is neglected, we
see that while the second example achieves a slightly smaller circular frozen orbit, it
has a cost of 268.9 km/s which is 5.5 times greater than the transfer via the first
example (see Table 4.1). Both circular orbits are stable there so is a clear advantage
90
to following the first example. Since the cost to achieve the stable circular orbit is
quite small (less than 50 m/s) it is possible that a low-thrust vehicle could achieve
this orbit. Following that, the low-thrust vehicle could transition down to a tighter
circular orbit with a spiral maneuver.
As another approach to transferring from a safe capture trajectory to a circular
frozen orbit, we examine the possibility of performing a simultaneous circularization
and plane change maneuver. It is not possible to do this maneuver with the current
example, since for this maneuver to be feasible the argument of periapsis at the
periapsis passage must be either 0o or 180o and for our current example the argument
of periapsis is approximately −90o. Therefore, we look back to our data of all
the periapsis passages of capture trajectories at this energy level, J = −1.60, and
examine their arguments of periapsis. Figure 4.17(a) shows argument of periapsis
plotted as a function of eccentricity for all of the periapsis passages of the capture
trajectories with J = −1.60. This plot includes periapsis passages with any radius
of periapsis. However, as before, our goal is a final orbit with a low altitude.
Therefore, restrict the periapsis passages to those with an altitude less than 250 km.
Figure 4.17(b) also shows the argument of periapsis as a function of eccentricity
for the periapsis passages that satisfy this condition. We see that the argument of
periapsis values lie between about 50 to 150o and −150 to −50o. Therefore, there
are no periapsis passages with altitudes close to the surface for which a simultaneous
circularization and plane change maneuver can be performed.
We conclude, from comparing the two plots in Figure 4.17, that the periapsis
passages with lower arguments of periapsis must have a large radius. We also see
that these periapsis passages have lower eccentricities. This is not surprising with the
doubly averaged dynamics in mind. In the doubly averaged system, the semi-major
axis is constant on average. Therefore, since the capture trajectories originate far
from the planetary satellite, for them to have a periapsis passage with a low altitude,
91
(a) (b)
Figure 4.17: ω as a function of eccentricity for safe capture trajectories with J = −1.60. (a) Allperiapsis passages. (b) Periapsis passages with altitude ≤ 250 km.
the eccentricity must be large. The numerical results support this since when all
periapsis passages are considered, they range in eccentricity from ∼ 0.32 → 0.75, but
when only those with altitudes less than 250 km are considered, the eccentricity only
ranges between about 0.705 → 0.75. Therefore, by limiting the radius of periapsis,
only periapsis passages with large eccentricities remain. Considering the averaged
dynamics once again, note that for a trajectory to reach a large eccentricity it must
be circulating about a libration region and libration regions are centered about
±90o. Then, examining Figure 4.17(b) it is clear this is the case. The arguments
of periapsis in this plot extend about ±50o on either sides of ±90o. We therefore
conclude that it is not dynamically possible to have a periapsis passage of a capture
trajectory that will allow a simultaneous circularization and plane change maneuver
to place the trajectory in a tightly bound circular orbit.
92
CHAPTER 5
THE DESIGN OF LONG LIFETIME SCIENCE
ORBITS
The requirements for a science orbit about a planetary satellite are generally a
near-polar, low-altitude orbit so that the science requirements of the mission, such
as imaging the surface and measuring various features of the planetary satellite,
are satisfied. Since the environment in the vicinity of many planetary satellites,
such as Europa, is highly perturbed, it is important for a science orbit to have a
long lifetime. For example, the nominal length of the science phase of a mission to
Europa could be one month, but it is essential to ensure that the spacecraft will not
impact with the planetary satellite if a particular control maneuver fails. Therefore,
the goal of long lifetime orbit design for Europa is lifetimes of around 100 days. This
will ensure that the spacecraft will be safe while awaiting the next possible maneuver
opportunity.
From Chapter 3, we have seen that stable frozen orbits exist at small inclinations
for low-altitude circular orbits (in the tide + J2 + C22 + J3 model) and at larger
altitudes (since the orbits are elliptic) for near-polar orbits (in tide-only and non
tide-only models). However, neither of these types of stable frozen orbits satisfy the
science requirements. Thus, we look towards unstable frozen orbits as a means for
designing long lifetime orbits that satisfy the science requirements. If a spacecraft is
arbitrarily placed in a near-polar, low-altitude orbit its lifetime is somewhat random
and can range from very short to very long. See, for example, Figure 5.1 which
93
0 20 40 60 80 100 120 140 160 1800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
time (days)ec
cent
ricity
impact
Figure 5.1: Eccentricity as a function of time for trajectories with an initial altitude of 100 km,initial inclination of 70o and different initial argument of periapsis values.
shows the eccentricity as a function of time for a few trajectories in the Europa
system. These trajectories have an initial altitude of 100 km, an inclination of 70o
and different initial argument of periapsis values. Their lifetimes vary from 40-165
days, demonstrating that arbitrarily placing a spacecraft in a low-altitude near-polar
orbit can result in impact after a relatively short period of time.
It will be shown later in this chapter that integrating a near-circular, near-polar
unstable frozen orbit in the 3-DOF system (including J2, C22 and J3) does not yield
a very long lifetime orbit. However, the structure of the dynamics in the vicinity
of these unstable frozen orbits helps in the design of long lifetime trajectories.
Specifically, since they have non-zero eccentricities, their stable and unstable
manifolds in the 1-DOF system are shifted and this causes variations in the rate of
change of the eccentricity along their paths. These variations can be exploited by
choosing the path that remains in the vicinity of a frozen orbit for the longest period
of time. If a trajectory in the full 3-DOF system follows this path, it can have a
relatively long lifetime on the order of at least 100 days, as compared to lifetimes as
short as 40 days for randomly chosen initial conditions.
After identifying the desired trajectory in the 1-DOF system, we develop a
94
method to systematically compute initial conditions in the 3-DOF system such that
the trajectory in that system will follow the 1-DOF system on average. To do this,
we use all three systems identified thus far, namely, the 1, 2 and 3-DOF systems.
We start with a point on the stable manifold of a frozen orbit in the 1-DOF system.
We then linearize the 1-DOF system about that point on the manifold and obtain
corrections such that the initial conditions lead to a trajectory in the 2-DOF system
that has motion in the 1-DOF system as its average. This procedure is repeated on
the 3-DOF system, by linearizing about the corrected point in the 2-DOF system.
The result is a set of initial conditions in the 3-DOF system that, when integrated,
follow the 1-DOF system on average.
A toolbox to compute long lifetime science orbits using the methods described in
this chapter was written in Matlab and delivered to the Jet Propulsion Laboratory
so they could use it to design these orbits. It is described at the end of this chapter.
5.1 Dynamics in the 1-DOF System
The frozen orbits that will be the basis of our analysis in this chapter are the
near-polar, near-circular unstable frozen orbits that exist in the tide + J2 + C22 +
J3 model. They are indicated by the heavy dashed line in Figure 5.2. Although
these frozen orbits have constant orbital elements on average in the 1-DOF system,
they are unstable, and so initializing at one in the 3-DOF system results in the
trajectory diverging from the frozen orbit location. Recall Figure 3.16, where we see
the trajectory in the (e,ω) plot diverge from its initial frozen orbit location. The
trajectory will impact with the planetary satellite after a short period of time since,
as seen in the plot, the eccentricity increases very rapidly leading to a decrease in
the radius of periapsis. Thus, the unstable frozen orbits can’t be used directly as
long lifetime orbits.
Since the 1-DOF system is Hamiltonian and the near-circular, near-polar frozen
1-DOF trajectory initialized at xm0 , both the 2-DOF and 1-DOF trajectories are
plotted. As Figure 5.7 shows, the dashed lines representing the 1-DOF trajectory
pass through the center of the solid curves which represent the 2-DOF trajectory.
5.2.2 From 2-DOF System to 3-DOF System
In this section, initial conditions in the 3-DOF system that will produce motion
that follows the 2-DOF system on average are computed. Since the results from the
previous section are incorporated, the resulting motion also follows the 1-DOF system
on average. In this case, there are six equations and x =
[a e i ω Ω σ
]T
where σ = M − nt and n is the mean motion of the spacecraft about the planetary
satellite. The time derivatives of these orbital elements are those stated in
106
Eqs.(3.4)-(3.7) plus the equation for σ taken from [8]:
da
dt=
2
na
∂R
∂M, (5.18)
de
dt=
1− e2
na2e
∂R
∂M−√
1− e2
na2e
∂R
∂ω, (5.19)
di
dt=
1√1− e2na2
(cot i
∂R
∂ω− csc i
∂R
∂Ω
), (5.20)
dω
dt=
√1− e2
na2e
∂R
∂e− cot i√
1− e2na2e
∂R
∂i, (5.21)
dΩ
dt=
1√1− e2na2 sin i
∂R
∂i, (5.22)
dσ
dt= − 2
na
∂R
∂a− 1− e2
na2e
∂R
∂e. (5.23)
For this system, the g0 components correspond to the LPE applied to R and the g1
components correspond to the LPE applied to R− R. Therefore, the g0 components
are time-invariant and the g1 components are time-periodic. In this case, the time
periodicity appears in terms of the true anomaly, ν, with a period of 2π which
corresponds to one period of the spacecraft about Europa. Since the true anomaly
is a function of the mean anomaly, the equations are expressed as derivatives with
respect to mean anomaly rather than with respect to time to simplify the analysis.
To transform from time derivatives to derivatives with respect to mean anomaly, the
equations must be multiplied by 1/n.
Since our goal is to determine initial conditions such that the 3-DOF trajectory
has the 2-DOF trajectory as its average, we linearize about a point in the 2-DOF
system. In particular, we linearize about the point in the 2-DOF system that
we obtained in the previous section and let this point be x0 = xm0 + δxm
0 . The
constant matrix A is still straightforward to compute analytically even though it is
a 6x6 matrix, and D is once again computed using the Romberg algorithm given
in Appendix A. However, the vector B, which in this case is B(ν(M)), cannot be
computed analytically since Kepler’s equation must be solved to determine the true
107
anomaly, ν, as a function of the mean anomaly, M . Therefore, we turn to the
computation of E. The computation of E where
E =1
2π
∫ 2π
0
∫ M
0
eA(M−τ)B(ν(τ))dτdM (5.24)
can still be accomplished using the double integral version of Gaussian quadrature
again, but the algorithm must be modified to solve Kepler’s equation, and hence
compute B at each step. This modified algorithm to compute E term-by-term is
also given in Appendix A. The corrections to x0, δx0, are then computed.
We now continue with the example from the previous section to show that when
integrated with the initial conditions obtained from the linearized system, the 3-DOF
system follows the 2-DOF system on average. The point that is linearized about,
which is also the initial condition for the 2-DOF system, is
x0 = xm0 + δxm
0 =
1682.53057218 km
0.02027136
70.860673330
171.18752273o
0o
0o
,
The corrections to these initial conditions, δx0 are found to be:
δx0 =
2.98938278 km
0.00219834
0.00239633o
0.17321674o
−0.00679549o
−0.07757915o
.
Figure 5.8 shows plots of the integrated 3-DOF trajectory, initialized at x0 +δx0, and
108
0 50 100 150 200 250 300 350 4001679
1680
1681
1682
1683
1684
1685
1686
sem
i−m
ajor
axi
s (k
m)
mean anomaly (deg)
0 50 100 150 200 250 300 350 4000.017
0.018
0.019
0.02
0.021
0.022
0.023
ecce
ntric
itymean anomaly (deg)
0 50 100 150 200 250 300 350 40070.81
70.82
70.83
70.84
70.85
70.86
70.87
70.88in
clin
atio
n (d
eg)
mean anomaly (deg)
0 50 100 150 200 250 300 350 400164
166
168
170
172
174
176
178
ω (
deg)
mean anomaly (deg)
Figure 5.8: Semi-major axis, eccentricity, inclination, and argument of periapsis as functions ofmean anomaly. Solid lines - 3-DOF trajectory, initialized at x0 + δx0. Dashed lines - 2-DOFtrajectory, initialized at x0.
the integrated 2-DOF trajectory, initialized at x0. In all four orbital element plots,
the 2-DOF trajectory cuts through the center of the 3-DOF trajectory. Therefore,
the goal of the 3-DOF motion following the 2-DOF motion on average has been
accomplished.
5.3 Long Lifetime Orbits in the 3-DOF System
Now that we have generated initial conditions in the 3-DOF system that follow
the manifold in the 1-DOF system on average, it is necessary to verify that these
initial conditions actually generate long lifetime orbits. Consider, for example,
Figure 5.9 which shows the integration, in the 3-DOF system, of the initial conditions
109
−0.08 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.08
−0.08
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
k=ecosω
h=es
inω
impact
Figure 5.9: Integration in the 3-DOF of a trajectory initialized at x0 with the 1-DOF manifolds.
generated in the previous section. Although it was shown in the previous section
that the 3-DOF motion did follow the 1-DOF on average, the trajectory will not, in
general, follow the 1-DOF manifold exactly since the linearization used is only an
approximation to the true motion. When several examples of potential long lifetime
orbits are computed, we find that sometimes the correct manifold path is followed
and sometimes it is not. In all of the cases, the trajectory does not have the desired
manifold path as its exact average. For the case in Figure 5.9, the trajectory follows
the wrong manifold path and has a lifetime of 84 days. This is not unexpected since
there should about a 50/50 chance of the trajectory lying on one side of the manifold
or the other.
To ensure that the trajectory always follows the desired manifold path a small
correction in the argument of periapsis, called the bias, is introduced. We show that
it is possible to find a particular argument of periapsis correction that gives the
maximum long lifetime orbit for a specific set of initial conditions. For the example
in Figure 5.9, the bias in the argument of periapsis that produces the longest lifetime
orbit is -0.0067 radians (−0.38o). This maximum lifetime orbit follows the 1-DOF
manifold exactly. However, from a practical mission standpoint, it is not that useful
110
0 20 40 60 80 100 120 140 160 1800
0.02
0.04
0.06
0.08
time (days)
ecce
ntric
ity
Figure 5.10: Time history of the eccentricity for a long lifetime orbit.
to find the maximum lifetime orbit. This is because the uncertainty in the orbit’s
position is, in general, greater than the accuracy with which the maximum lifetime
orbit is determined. In addition, since the maximum lifetime orbit is one that follows
the 1-DOF manifold very closely, a very small error in the initial position could
push the trajectory to the wrong manifold path. The details regarding the orbit
uncertainty will be discussed in the next chapter. For now, we focus on finding long
lifetime orbits that follow the desired manifold path, but do not necessarily have
maximum lifetimes.
5.3.1 Minimum Eccentricity on the Manifold
In the example that has been discussed thus far, the initial eccentricity on the
manifold was chosen arbitrarily to be 0.02 since the only requirement for an initial
eccentricity is that it be on the stable manifold of the long lifetime orbit. There
is however, a more systematic way to choose the initial eccentricity. Consider
Figure 5.10 which shows the time history of the eccentricity for a long lifetime orbit
corresponding to a 70o inclination frozen orbit. Note in this plot that the eccentricity
decreases rapidly before it settles around the frozen orbit value where it stays almost
constant for the majority of the orbit. Observe further that just before the constant
value is reached, the eccentricity goes through a minimum. It is this minimum
eccentricity value that will be used from now on to initialize long lifetime orbits.
One advantage of using this minimum eccentricity value is that the period of time
111
−0.03 −0.02 −0.01 0 0.01 0.02 0.03−0.03
−0.02
−0.01
0
0.01
0.02
0.03
k=ecosω
h=es
inω
emin
frozenorbit
manifoldpoint
Figure 5.11: 1-DOF manifolds with minimum eccentricity circle (dashed curve).
before the trajectory reaches this value is quite short compared the amount of time
it spends oscillating about the frozen orbit. Thus, not much orbital lifetime is gained
by starting at a higher eccentricity value. In addition, always initializing trajectories
at the minimum eccentricity for the frozen orbit under consideration provides a good
way to compare the lifetimes of the orbits. Finally, in terms of controlling the orbit,
initializing at the minimum eccentricity provides advantages that will be shown in
the following chapter.
The minimum eccentricity on the stable manifold can be computed geometrically.
First, consider Figure 5.11. The line connecting the frozen orbit point to the
manifold point is the linear approximation of the manifold in that region. Although
the manifold in the 1-DOF system is governed by a very complicated function, it can
be approximated by a line in the vicinity of the frozen orbit, as seen in Figure 5.3(b).
The manifold point is a point that has the same eccentricity as the frozen orbit and
an argument of periapsis consistent with the 1-DOF manifold (computed using the
method in section 5.2). The minimum eccentricity then lies on this line and is the
point with the minimum distance from the origin, as shown in Figure 5.11. Let the
frozen orbit point be (k∗,h∗) and the manifold point be (km,hm). Then, the equation
112
for the line connecting the two points is:
h =
(hm − h∗
km − k∗
)(k + km)− hm , (5.25)
and the minimum distance from the origin to this line is[43]:
d =
∣∣∣∣∣∣∣det
km − k∗ k∗
hm − h∗ h∗
∣∣∣∣∣∣∣√
(hm − h∗)2 + (km − k∗)2. (5.26)
Since, in the (k,h) plot, the eccentricity is equivalent to the radial distance from the
origin to a point, the minimum eccentricity is given by the distance d in Eq.(5.26).
Then, the corresponding argument of periapsis on the manifold can be computed
by the method in section 3.2. Note that since the line used to the approximate the
manifold is not an exact solution of the manifold, it is only used to compute the
minimum eccentricity (distance to the line) and not the argument of periapsis (angle
between the origin and the line). Thus, the (e,ω) pair actually sits on the manifold.
If the approximation was used to compute both values, the resulting pair would not
lie on the manifold, but on the approximation to the manifold.
5.3.2 Examples of Long Lifetime Orbits
The general algorithm used to compute long lifetime orbits is given in Table 5.1.
As previously noted, it is not always necessary to find the maximum lifetime orbit
corresponding to a particular frozen orbit. This maximum lifetime orbit will be the
trajectory that follows the desired manifold path exactly (i.e. it has the manifold
as its average). The goal here is to show that it is possible to design long lifetime
trajectories that follow in general the II→I path as in Figure 5.4. They do not need
to have the manifold has their exact average, but must lie relatively close to the
manifold. The first examples presented are for the Europa system, with the tide, J2,
C22 and J3 components included in the model. Following that, we also show that it
113
Table 5.1: Algorithm for computing a long lifetime orbit
1. Choose frozen orbit in the 1-DOF system based on its inclination.2. Compute the minimum eccentricity point on the stable manifold of the
frozen orbit.3. Compute the corrected initial conditions to the 2-DOF system by using the
linearization of the 2-DOF system about the point on the manifold.4. Compute the corrected initial conditions to the 3-DOF system by using the
linearization of the 3-DOF system about the corrected 2-DOF initial conditions.5. Find an argument of periapsis value that produces a long lifetime orbit by
numerical search, if necessary.
is possible to design orbits for other planetary satellites such as the Saturnian moons
Enceladus and Dione. Since, as demonstrated in Chapter 3, the environments in the
vicinity of those moons are much more unstable than the Europa environment, the
long lifetime orbits designed for those systems are not nearly as long-lived as those
in the Europa system.
Starting with the Europa system, trajectories are computed using the minimum
eccentricity point on the manifold in step 2 of the algorithm in Table 5.1. We find
that the trajectory follows the desired manifold path without a bias correction in all
the cases, which cover the entire range of unstable, near-polar, near-circular frozen
orbits. The lifetimes of these trajectories are:
Frozen Orbit Lifetime
Inclination (days)
70o 105
75o 119
85o 110
95o 96
105o 95
110o 99
Note that the lifetimes of the direct orbits are all larger than the lifetimes of
114
the retrograde orbits. It has been shown that direct and retrograde orbits have
different properties in the 3-DOF system and that this difference disappears during
the averaging technique used here [22]. Therefore, it is possible that, due to the
averaging used for this analysis, the initial condition computation algorithm provides
better results for direct than for retrograde orbits. These trajectories fulfill the goal
of designing a long lifetime orbit that follows the desired manifold path, without
having it as an exact average. It is an interesting exercise to demonstrate that the
algorithm brings us very close to a trajectory that follows the manifold exactly (i.e.
the maximum lifetime orbit). To find the maximum lifetime orbit for each case, a
numerical search over argument of periapsis values is performed to find the necessary
bias (step 5 of the algorithm in Table 5.1). Also, note that although all of the
trajectories initialized at the minimum eccentricity point follow the correct manifold
path in the Europa case, as shown above, that will not necessarily be true for all
initial points on the manifold. For those cases, a numerical search over ω values can
be performed to either find the maximum lifetime orbit or any long lifetime orbit
that follows the correct manifold path.
Table 5.2 shows the bias in the argument of periapsis that produces the longest
lifetime orbit to an accuracy of 0.0002 radians (0.011o) for each frozen orbit
inclination case discussed above. Note that all of the bias values are less than about
one and a half degrees and the lifetimes increase by between 45 and 90 percent. This
large variation in improvement in lifetime occurs since the maximum lifetime orbit is
only being computed to an accuracy of 0.0002 radians in the argument of periapsis
bias. If a smaller step size in argument of periapsis was used, the lifetimes would all
increase by percentages in the higher range. Also note that the bias values are larger
for the retrograde orbits than for the direct orbits. This is probably because the
original lifetimes of the retrograde orbits, using the unbiased value of ω, are smaller
than the original lifetimes for the direct orbits. Although the existence of these
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Table 5.2: Long Lifetime Orbits for the Europa System
Frozen Orbit Unbiased Lifetime Maximum Lifetime Bias (rad)Inclination (days) (days)
Using the results in [40] as a guide, we study how errors in the spacecraft’s
position affects the designed long lifetime orbits. To study the orbit uncertainty
of long lifetime orbits, Monte Carlo simulations are performed. A large number of
trajectories are integrated, each with a random initial position error. The initial
position error is randomly determining for radial, crosstrack and alongtrack errors
using a Gaussian distribution with a particular standard deviation. The first set
of results are for 3-sigma values corresponding to the realistic gravity error noted
above. Each of the six long lifetime orbits with different inclinations from section
5.3.2 are considered, with 200 trajectories integrated for each. Recall that the six
long lifetime orbits are initialized at the minimum eccentricity point and follow the
correct unstable manifold but are not the maximum lifetime orbits for each case.
Figure 6.14 shows the Monte Carlo simulation results for the minimum
eccentricity long lifetime orbits. Each cluster contains several points, each of which
represents the stopping point of a trajectory integrated to determine which manifold
path it follows. Observe that in four of the six cases, all of the 200 trajectories follow
the correct manifold path. For the 75o inclination case, 44 of 200 trajectories (22%)
follow the wrong manifold path and for the 85o inclination case, 15 of 200 trajectories
(7.5%) follow the wrong manifold path. These results are explained by taking into
account the lifetimes of the six original long lifetime trajectories (Table 5.2). The
two trajectories with the longest lifetimes are the 75o and 85o inclination cases. A
longer lifetime means that the trajectory follows the 1-DOF manifold more closely.
The shorter lifetimes that follow the correct manifold path therefore correspond
the trajectories that lie slightly above the manifold. Then, the more closely the
correct manifold path is followed, the more likely it is that an initial position error
will cause the trajectory to follow the wrong manifold path. Since the 75o and 85o
146
Table 6.2: Monte Carlo Results for Long Lifetime Orbits
Frozen Orbit Original Orbit Maximum OrbitInclination Lifetime Correct % Lifetime Correct %
70o 105 days 100 182 days 47.275o 119 days 78.0 174 days 50.585o 110 days 92.5 171 days 52.595o 96 days 100 182 days 56.5105o 95 days 100 172 days 51.0110o 99 days 100 188 days 48.0
inclination orbits have the longest lifetimes, they follow the 1-DOF manifold more
closely and therefore it is not surprising that in some cases an initial position error
results in a trajectory that follows the wrong manifold path. Further, note that the
75o inclination orbit has a longer lifetime than the 85o inclination orbit. It follows
then that a larger percentage of the trajectories in the 75o case follow the wrong
path than the 85o case.
The theory that the closer a long lifetime trajectory is to the 1-DOF manifold,
the more susceptible it is to errors in the initial position can be further verified by
performing Monte Carlo analysis of some maximum long lifetime trajectories. The
same maximum lifetime trajectories as those in Table 5.2 are used, and 200 Monte
Carlo simulations are run for each case. The initial position errors are once again
determined randomly via a Gaussian distribution with the same 3-sigma values as
above. Table 6.2 shows the results, along with the lifetimes of the maximum lifetime
orbits. Observe that the number of trajectories that follow the correct manifold is
approximately 50% in each case. This is exactly what is expected since the maximum
lifetime trajectories follow the 1-DOF manifolds very closely. Therefore, a small
initial position error can bump the trajectory either just above the manifold or just
below the manifold. If it is just above the manifold, it will follow the correct manifold
path and if it is just below the manifold, it will follow the incorrect manifold path.
It is important when planning a mission to a planetary satellite such as Europa
Figure 6.14: Monte Carlo simulation results for various long lifetime trajectories (200 trajectoriesfor each case). Each point in the plot represents the stopping point of a trajectory once it reachesthe desired (top right) or not desired (bottom left) manifold.
148
to ensure that the uncertainty error in the position of the spacecraft will not lead
to impact with the planetary satellite. Therefore, using the maximum lifetime orbit
as the nominal science orbit is not the best course of action since there is about
a 50% chance that the spacecraft will deviate too far from the nominal trajectory.
The further above the manifold the long lifetime trajectory is, the more likely it
is to follow the correct manifold path. However, the farther above the manifold
the trajectory is, the shorter lifetime it has. Choosing the nominal trajectory is
therefore a trade off between guaranteeing that it will follow the correct manifold
path and having a long lifetime. In addition, before the mission starts, the orbit
accuracy possible will not be known exactly. The values used here are based on the
assumption that prior to the science portion of the mission the gravity field will be
determined with some level of certainty. Even if a trajectory is chosen such that it
will always follow the correct manifold path for the error levels used here, it is not
certain that this level of orbit accuracy is attainable. Therefore, we discuss in the
next section how to return the spacecraft to its nominal orbit once it has drifted
away due to an initial error.
6.4 Long Lifetime Orbits with Initial Errors
It is inevitable that during the science phase of a mission to a planetary satellite that
the spacecraft will drift away from the nominal trajectory. This will most likely be
due to initial errors in the spacecraft’s position and velocity. Once it is determined
that the spacecraft is no longer on the nominal trajectory, it is important to devise
a method for returning it to the nominal trajectory so that the mission can proceed
as planned. For example, recall in section 6.1 the method to reset the long lifetime
orbit once the spacecraft reaches the unstable manifold. As shown in section 6.3, due
to initial position errors, it is possible that a long lifetime trajectory will follow the
wrong manifold path. If it does so, it is not possible to use the algorithm in section
149
6.1 to reset the trajectory. Therefore, it is important to return to spacecraft to its
nominal trajectory before it has the opportunity to follow the wrong manifold path.
6.4.1 Design of the Correction Maneuvers
The original long lifetime orbit under consideration is determined as usual from the
minimum eccentricity point. Initial errors are introduced by randomly determining
position errors as in the previous section. A time (in days) is chosen to represent the
amount of time until the correction to the trajectory will be made. The amount of
time (TE) is long enough for the spacecraft to drift away from the nominal trajectory.
The maneuvers to be used for the correction are two burns corresponding to a
Hohmann-type transfer. Therefore, the initial transfer point should be at periapsis
and the target at apoapsis. The initial transfer point is determined by integrating
the erroneous trajectory until the closest periapsis passage to TE (either before or
after). Its initial position and velocity are denoted as r0 and v0, respectively. In
addition, denote its initial longitude of the ascending node as Ω0.
The nominal trajectory is the long lifetime orbit with no initial errors. The goal
of this correction scheme is to transfer the spacecraft to the nominal trajectory. The
difficulty lies in determining how to designate the target on the nominal trajectory.
Unfortunately, it is not as simple as integrating the nominal trajectory for the
same amount of time as the erroneous trajectory and setting the target accordingly.
Using this method creates inconsistencies between the initial and target longitude of
ascending node values which creates difficulties in converging to the transfer orbit.
The method that produces the best results is to integrate the nominal trajectory
until its longitude of ascending node value is equal to Ω0 and then finding the
closest periapsis passage to that point (before or after). This periapsis passage on
the nominal trajectory therefore corresponds to point where the spacecraft on the
erroneous trajectory should be. Then, the target is set to be the first apoapsis
150
Figure 6.15: Schematic of Hohmann transfer between two elliptic orbits.
passage following that periapsis passage on the nominal trajectory. Denote the
target position and velocity as rT and vT , respectively.
As in section 6.2, the goal of the transfer is to reach the target and minimize
the sum of the squares of the costs of the two burns (Eq.(6.33)). The exact same
algorithm is used, with the only change being the ideal, first guess transfer orbit.
The first guess for the first maneuver is an eccentricity change at periapsis such
that the transfer orbit arrives at the target at periapsis. Since, as before, there
are perturbations in the system that prevent the target from being reached by this
simple method, it is used as a first guess and corrections are made. Figure 6.15 is a
diagram of a Hohmann transfer between two elliptic orbits. Let the semi-major axis
of the transfer orbit, atr be given by
atr =r0 + rt
2. (6.42)
Then, the velocity of the first guess transfer orbit is:
vg =
õ
(2
r0
− 1
atr
)v0 , (6.43)
where v0 is the direction of the initial velocity v0.
151
The overall procedure to compute the maneuvers necessary to return the
spacecraft to its nominal trajectory mirrors the procedure in Section 6.2 and is as
follows:
1. Compute the ideal transfer consisting of an eccentricity change at periapsis
where the velocity is vg and integrate to apoapsis. Let the time required for
this transfer be tf .
2. Use the iteration scheme
δv = δv0 + δv1 + δv2 + · · · , (6.44)
where
δvi+1 = φ−1rv δri (6.45)
to compute the necessary correction to the first burn such that δr = 0.
3. Let ∆v∗1 = δv and ∆v∗2 = vt − v(tf ) and compute new corrections to the
burns, δv1 (Eq.(6.31)) and δv2 (Eq.(6.32)), to minimize the cost using δt = −BA
(Eqs.(6.38),(6.39)).
4. Repeat steps 2 and 3 until δti+1 − δti < ε where ε = 10−9 for our simulations.
6.4.2 Results
The algorithm described above is designed to correct the position and velocity of
a spacecraft after it has drifted away from its nominal trajectory due to an initial
error. The question is, for how long after the initial error is it possible to correct
the position? After a certain amount of time, the correction algorithm will not
converge to a transfer orbit with maneuvers that have acceptable costs. The longer
a spacecraft follows an erroneous trajectory, the more difficult is it to correct.
Therefore, after a certain point, as long as the spacecraft follows the correct manifold
152
path, it makes more sense to reset the orbit once it reaches the unstable manifold,
as in section 6.1, instead of using the technique described in this section.
The first set of examples presented have initial errors that follow the realistic
model of gravity field uncertainty described in [40]. For each long lifetime orbit case,
random initial position errors in the radial, crosstrack and alongtrack directions
are introduced, and the trajectory is integrated for 1, 5, 10 and 20 days before the
correction is made. For each case, 40 simulations are performed, and the average
cost for each maneuver is computed. Corrections are not made after periods of time
longer than 20 days since after that length of time, the trajectory has drifted far
enough away from the nominal trajectory such that this method is not feasible. This
is likely because the method is based on an initial guess which is the ideal transfer
occurring in the absence of all perturbations. It also assumes that the initial and final
orbits are co-planar which is, in general, not the case. However for corrections done
within 20 days of the initial error, these assumptions are valid and the algorithm
converges to a transfer orbit with low-cost maneuvers. After the trajectory has
evolved for longer, the algorithm does not always converge to a transfer orbit, and
when it does, the costs are sometimes very large. Therefore, this method is only
valid when the spacecraft has not drifted too far from the nominal trajectory.
The resulting costs for the two maneuvers with the error model described above
are shown in Table 6.3. The factors to be considered when determining when the
correction should be made are: how quickly after discovering that the trajectory is
not on the nominal trajectory the correction can be performed and when it is most
cost-effective to perform the correction. In examining Table 6.3, we see that the
lowest cost correction occurs at the 1 day mark. However, depending on when it
becomes known that the spacecraft is not on the nominal trajectory, it might not be
possible to perform the maneuver after 1 day. Therefore, the fact that the cost is
still relatively low for corrections after 5 and 10 days is a useful result showing that
153
Table 6.3: Average cost to correct a long lifetime trajectory after an initial position error
1 day 5 days 10 days 20 daysFrozen Orbit ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2
Inclination m/s m/s m/s m/s70o 0.0989 0.1441 0.3733 0.544985o 0.2190 0.3589 0.8830 2.388595o 0.1642 0.2986 0.6010 3.7737105o 0.1547 0.2949 0.3299 1.8024
it is not necessary to perform the correction very close to the time when the error is
discovered. These results show that performing a correction maneuver within about
a week or two from the time of the initial error is feasible.
The distribution used to compute initial errors for the results in Table 6.3 are
those designated in [40] as the most likely orbit uncertainty values for the science
portion of a mission to Europa. As previously noted, these orbit uncertainty values
were obtained by assuming that prior to the science phase of the mission, the
gravity field of Europa could be determined more accurately than the currently
available values. Therefore, it is important to study the effect of larger levels of
orbit uncertainty on the ability to make corrections to the spacecraft’s position.
Table 6.4 shows results corresponding to initial error values double the size of those
in Table 6.3. In other words, the standard deviations of the error distributions
are two times the values in [40]. It is not surprising that an increase in the initial
position error leads to a decrease in the length of time that can elapse for a correction
to be possible. For the results in Table 6.4, attempts to do a correction after 20
days caused errors in the convergence of the transfer orbit, and large costs when a
transfer orbit was found. Therefore, the results shown are for corrections after 1 day,
5 days, 10 days and 15 days. First, note that the total cost to return the spacecraft
to its nominal trajectory is larger in this case than for the results in Table 6.3.
This is not surprising since the initial errors are larger in this case, and so larger
154
Table 6.4: Average cost to correct a long lifetime trajectory after a larger initial position error
1 day 5 days 10 days 15 daysFrozen Orbit ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2 ∆v1 + ∆v2
Inclination m/s m/s m/s m/s70o 0.2256 0.2466 0.8433 0.796685o 0.3746 0.7773 1.4339 1.329495o 0.4414 0.6415 1.3459 2.4245105o 0.2733 0.3783 0.7837 1.8184
maneuvers are needed. However, the costs for the corrections after 1 day, 5 days and
sometimes 10 days are still relatively low, and corrections are feasible once again
even if it is necessary to wait for a few days after the initial error. The results for
both error models are consistent since as time goes on, the deviation between the
erroneous trajectory and the nominal trajectory gets larger, increasing the cost of
the correction in the same way that a larger initial error also increases the cost of
the correction.
155
CHAPTER 7
CONCLUSIONS AND FUTURE DIRECTIONS
The main focus of this dissertation is to develop an understanding of spacecraft
dynamics in the vicinity of planetary satellites and to use these dynamics for mission
design. The main contributions are:
• The analysis of the 1-DOF system dynamics including the frozen orbits, their
stability and their dependence on the higher order gravity field terms of the
planetary satellite
• The analysis of capture trajectories in safe zones that allow an uncontrolled
spacecraft to remain in orbit and from which transfers to lower altitude orbits
are possible
• The design method for long lifetime science orbits using the stable and unstable
manifolds of the frozen orbits
• The optimal transfer strategy for restarting a long lifetime orbit using the fact
that the manifolds are offset from the origin
• The demonstration that it is possible, with relatively low costs, to return the
spacecraft to its nominal trajectory after initial position errors
156
7.1 Summary of the Results
The first main result obtained is the derivation of frozen orbit solutions in the
1-DOF system that are relevant for planetary satellites. When the J3 planetary
satellite gravity field term is included in the model, the structure of the frozen
orbit solutions change, qualitatively independent of the size of J3. Therefore, this
analysis is applicable to any planetary satellite system that can be approximated by
the Hill problem, no matter what its J3 value is. Both stable and unstable frozen
orbit solutions exist, and the stable frozen orbit solutions are long-term stable in
the 3-DOF system as long as their altitude is not too large. These stable orbits are
not of interest from a scientific perspective, however, due to their low inclinations.
The unstable frozen orbit solutions are of interest to scientific observations due to
their high (near-polar) inclinations. Thus these orbits are used to motivate science
orbit design. These are especially challenging as an uncontrolled, or poorly designed,
science orbit will impact on the planetary satellite surface in a relatively short time.
The investigation of capture trajectories in the Hill 3-body problem is
accomplished using a Periapsis Poincare map and a specific application is made to
trajectories about Jupiter’s moon Europa. However, the same analysis could be
repeated for other planetary satellites of interest. Trajectories that don’t impact
with Europa or escape for one week time periods are identified and the regions in
which they occur denoted as safe zones. These regions are characterized by the first
periapsis of each trajectory. It is an important result that the determination that
a trajectory will be safe for one week can be made from its first periapsis passage
location. These safe zones are evaluated to find trajectories from which it is possible
to transfer to long-term stable orbits with a relatively low cost. Orbits considered
as possible targets are both elliptic and circular frozen orbits which are either
long-term stable or can be stabilized with small control maneuvers. We found that
the lowest-cost method to transfer to a circular orbit is by using the dynamics of the
157
system to decrease the eccentricity rather than circularizing the orbit directly. In
particular we describe a low cost sequence that results in a circular orbit reachable
by a low-thrust spacecraft. From this orbit it could then spiral down into a lower
altitude circular orbit.
The design of science orbits with long lifetimes begins with the identification of
the stable and unstable manifolds of the low-altitude, near-polar unstable frozen
orbits in the 1-DOF system. The stable to unstable manifold path that has the
largest change in argument of periapsis while the orbit stays near-circular is identified
as the desired path for a long lifetime orbit. Following that, an algorithm is derived
that allows us to relate initial conditions on the 1-DOF manifold to initial conditions
in the 3-DOF system with an intermediate step in the 2-DOF system. This
algorithm allows for the semi-analytic computation of the desired initial conditions,
and easily incorporates changes in the system parameters. In addition, it utilizes the
main dynamical features of the system to extend the lifetimes as compared to the
unstable frozen orbits. The use of this method for Europa results in science orbits
with lifetimes of at least 100 days. Therefore, the use of this orbit protects the
spacecraft in the case of control maneuver failures. This method is also applied to
Saturn’s moons Enceladus and Dione to show that it is applicable to other planetary
satellites. Viable science orbits are designed in those cases, but with shorter lifetimes
since their environments are more highly perturbed. A toolbox for computing initial
conditions for long lifetime science orbits that was written and delivered to the Jet
Propulsion Laboratory is also described.
The control of long lifetime science orbits has been investigated for a few different
scenarios. The first attempt maintains constant orbital elements using continuous
thrust. However, the thrust levels required for this type of control are found to be
too large to be practical for a mission. Next, scenarios where a long lifetime science
orbit reaches the unstable manifold and needs to be reset to the stable manifold
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is studied. For this scenario, an algorithm to redesign a long lifetime orbit based
on the parameters of the trajectory at the unstable manifold is derived. Then, a
two-maneuver sequence is computed to execute a transfer to the new transfer orbit.
The total cost of these maneuvers is optimized, and fuel costs of around 30 cm/s/day
are obtained. Next, the effect of orbit uncertainty on the long lifetime science
orbits is investigated using Monte Carlo simulations. We found that the closer the
trajectory is to the 1-DOF manifold, the more susceptible it is to veering off the
desired manifold path due to an initial position error. Finally, a scheme is developed
to return the spacecraft to its nominal trajectory following an initial position error.
A transfer orbit is optimized by the total cost of the two maneuvers necessary, and
comparisons are made based on how many days after the initial error the correction
is executed. We found that even if it is necessary to wait 5 days after the initial
error, the cost to return the spacecraft to the nominal trajectory is less than 0.4 m/s.
7.2 Future Directions
In this section, a few possible extensions of the research described in this dissertation
are discussed.
7.2.1 Transfers from Capture Trajectories
In our discussion of safe zones for captured trajectories, the safe zones are identified
by their energy level (i.e. their Jacobi integral value). Transfers from safe capture
trajectories to stable frozen orbits are then computed using impulsive maneuvers
whereby the entire set of trajectories remain at the same energy level. An extension
of this work is to derive low-cost transfers that traverse safe zones at different energy
levels rather than staying within one energy level. This is of particular interest since
low-thrust trajectories are becoming more desirable with the increased use of electric
propulsion in spacecraft. In addition, rather than stopping the scheme at a stable
159
frozen orbit at a higher altitude, the low-thrust trajectory could be continued using
a spiraling-in approach to transfer the spacecraft to a low-altitude science orbit.
7.2.2 Science Orbit Design
The most logical extension for the science orbit design is to incorporate the science
orbits designed in this dissertation into precision navigation and mission design
programs that include detailed models of all of the physics acting on the system.
This is probably the most effective way to design science orbits for a real mission
since the advantage of our orbit design scheme is its semi-analytic results that are
only possible since the system is not too complex. However, since the system we use
is an approximation to the true system, at some point the more complicated true
system must be incorporated into the orbit design process.
7.2.3 Control of Long Lifetime Science Orbits
There are many issues that could be explored regarding the control of long lifetime
science orbits. Perhaps the most important is ensuring that the spacecraft remains
on the nominal trajectory. We have shown that it is possible to return the spacecraft
to the nominal trajectory after a few days following an initial position error. A more
robust technique would be to devise active controls such that the spacecraft always
stays on the nominal trajectory. This is particularly important in cases where the
system is so unstable that even leaving the trajectory uncontrolled for one day would
be disastrous. One such example is Enceladus, where we showed that long lifetime
science orbits have lifetimes of only 10 days.
Another possible extension concerns resetting the long lifetime science orbit
before impact with the planetary satellite. The transfer method described in this
dissertation involves computing the target (initial conditions of a new long lifetime
orbit) and then optimizing the transfer over the total cost. An improvement to
this method would have the optimization take place over both the total cost and
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the target itself. The goal of this resetting scheme is to transfer the spacecraft to
another long lifetime trajectory, but the exact location where the spacecraft ends
up on the new trajectory is not important. Therefore an optimization procedure
that allowed for variations of the target on the desired trajectory would most likely
produce lower-cost maneuvers than the current one which sets the target before the
optimization takes place.
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APPENDIX
162
APPENDIX A
NUMERICAL ALGORITHMS
This appendix details the algorithms used for the numerical computations of
initial conditions for science orbits in Chapter 5.
A.1 Romberg Algorithm
The number of iterations performed in this algorithm is defined by a number
n. Larger values of n correspond to a higher accuracy in the result, but also
longer computation times since more computations are being performed. We chose
n = 10 since we found that that value gives us a good estimate while keeping the
computation time not too large.
Purpose: Approximate the integral Ij,k which is the entry in the jth row and kth
column of:
I =
∫ b
a
exp (At)dt
1. h = b− a
2. R1,1 = [exp (Aa)]j,k + [exp (Ab)]j,k/2h
3. for i=2 to n
(a) sum = 0
(b) m = 2(i−2)
(c) for k=1 to m
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i. sum = sum + [exp (A(a + (k − 0.5)h))]j,k
(d) R2,1 = (R1,1 + h ∗ sum)/2
(e) for j=2 to i
i. L = 22(j−1))
ii. R2,j = R2,j−1 + (R2,j−1 −R1,j−1)/(L− 1)
(f) h = h/2
(g) for j=1 to i
i. R1,j = R2,j
4. Ij,k = R2,n
A.2 Gaussian Quadrature
For the Gaussian algorithm for double integrals, a set of weights and abscissas
must be specified. The values that we use[1] are denoted rj,k and coj,k, respectively.
The number of iterations performed is given by two numbers n and m. We chose
n = m = 12, which gives us a good approximation of the integral while ensuring not
too long of a computation time.
Purpose: Approximate the integral Ik which is the kth entry of:
I =
∫ b
a
∫ τ
0
exp (A(t− τ))B(τ)dτdt
1. h1 = (b− a)/2
2. h2 = (b + a)/2
3. aj = 0
4. for i=1 to m
(a) t = h1 ∗ rm−1,i + h2
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(b) jx = 0
(c) c1 = 0
(d) d1 = t
(e) k1 = (d1− c1)/2
(f) k2 = (d1 + c1)/2
(g) for j=1 to n
i. τ = k1 ∗ rn−1,j + k2
ii. q = [exp (A(t− τ))B]k
iii. jx = jx + con−1,jq
(h) aj = aj + com−1,ik1 ∗ jx
5. aj = aj ∗ h1
6. Ik = aj
A.3 Gaussian Quadrature with Kepler Equation
Modification
For the Gaussian algorithm for double integrals, a set of weights and abscissas
must be specified. The values that we use[1] are denoted rj,k and coj,k, respectively.
The number of iterations performed is given by two numbers n and m. We chose
n = m = 10 in this case, which gives us a good approximation of the integral while
ensuring not too long of a computation time. The number of iterations is smaller
than in Appendix B since this algorithm is used on a much more complex system
and increasing the number of iterations any larger than 10 resulted in prohibitively
long run times.
The function kepler(τ, e) computes the true anomaly ν given a mean anomaly τ
and an eccentricity e. This is done by using the built-in Matlab function ‘solve’ to
165
find the roots of
τ − E + e sin E = 0 (A.1)
where E is the eccentric anomaly. Then,
ν = cos−1 ((cos(E)− e)/(1− e cos(E))) (A.2)
Purpose: Approximate the integral Ik which is the kth entry of:
I =1
2π
∫ 2π
0
∫ M
0
eA(M−τ)B(ν(τ))dτdM
1. h1 = (b− a)/2
2. h2 = (b + a)/2
3. aj = 0
4. for i=1 to m
(a) M = h1 ∗ rm−1,i + h2
(b) jx = 0
(c) c1 = 0
(d) d1 = M
(e) k1 = (d1− c1)/2
(f) k2 = (d1 + c1)/2
(g) for j=1 to n
i. τ = k1 ∗ rn−1,j + k2
ii. ν = kepler(τ, e)
iii. q = [exp (A(M − τ))B(ν)]k
iv. jx = jx + con−1,jq
(h) aj = aj + com−1,ik1 ∗ jx
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5. aj = aj ∗ h1
6. Ik = aj
167
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ABSTRACT
ORBIT DESIGN AND CONTROL OF PLANETARY SATELLITE
ORBITERS IN THE HILL 3-BODY PROBLEM
by
Marci Paskowitz Possner
Chair: Daniel J. Scheeres
The exploration of planetary satellites by robotic spacecraft is currently of strong
scientific interest. However, sending a spacecraft to a planetary satellite can be
challenging due to strong perturbations from the central planet. The primary goal of
this dissertation is to identify and utilize the main dynamical features of the system
in the orbit design process.
The system is modeled using a modified form of Hill’s 3-body problem, where
the effect of the planetary satellite’s gravity field is included in the low-altitude
analysis. A thorough study of the dynamics of the system is performed by applying
averaging theory to reduce the complexity and degrees of freedom of the system.
The reduced system has one degree of freedom (DOF) and has equilibrium solutions
called frozen orbits. These frozen orbits are first used as targets for transfers from
capture trajectories in ‘safe zones’. The ‘safe zones’ in phase space are numerically
determined; they contain trajectories that enter the Hill region and allow an
uncontrolled spacecraft to remain in orbit without impact or escape for specified
time periods. Transfers from safe trajectories to frozen orbits are identified and
criteria on their costs evaluated.
Unstable low-altitude, near-polar frozen orbits are the basis for the design of long
lifetime science orbits. The stable and unstable manifolds of these frozen orbits in
the 1-DOF system are investigated and the desired path for long lifetime orbits is
identified. An algorithm is developed to systematically compute initial conditions in
the full system such that the orbits follow the desired path and have sufficiently long
lifetimes to be practical as science orbits about planetary satellites. The analysis of
the control of a planetary satellite orbiter begins with the evaluation of the effect
of orbit uncertainty on the science orbits and the identification of criteria to ensure
that the orbits have the desired behavior. Then, two control schemes are developed:
a) given the terminal conditions of a science orbit, redesign a new science orbit and
execute a low-cost transfer to it, b) return the spacecraft to its nominal trajectory