AAS 13-789
CONNECTING HALO ORBITS TO SCIENCE ORBITS AT
PLANETARY MOONS
Kevin A. Bokelmann* and Ryan P. Russell†
Increasing interest in planetary moons such as Europa calls for capturing into
low-altitude science orbits. In this study, the maneuver costs for this final phase
are investigated. Analytic expressions are developed to predict the minimum velocity change to transfer between loosely captured orbits and low-altitude science orbits at different energy levels. The equations are derived in both the restricted three-body problem and the Hill’s approximation. Visualizations of the
results are developed to allow rapid assessment of transfer propulsive costs of
interest. Multiple impulsive transfers are calculated to validate the theory.
INTRODUCTION
The recent increased interest in the exploration of the outer planets has led to several studies
examining missions to the Jovian and Saturnian systems. Several studies focus on the use of the
low-energy dynamics of the three-body problem to reduce propulsive costs from typical two-body
sphere of influence models. Starting with the inter-moon phase, Lantoine et al. investigated connecting two moons using three-body halo orbits as boundary conditions using a combination of
multi-body and patched three-body dynamics.1 Their results and others show that such transfers
can be completed with relatively little
(on the order of tens of meters per second).2 Related
work by Nakamiya et al. focuses on capture into these periodic orbits from more general interplanetary transfers, showing reduced costs from typical direct methods.3
Once captured into periodic orbits several mission design options are available. The most scientifically compelling option is to ultimately transfer into a low-altitude science orbit. Baoyin et
al. looked into transfers from the Lagrange points and Lyapunov orbits that provide wholesurface coverage of both primaries using numerical integration and invariant manifolds of the
three-body dynamics.4 A ballistic lunar trajectory from the Earth to the Earth-Moon L2 was modeled by Parker et al. using dynamical systems theory where they show significant savings over
direct trans-lunar trajectories.5 Further work by Russell and Lam used stable manifolds of unstable periodic orbits to generate ballistic capture trajectories to highly-inclined science orbits before
transition to a low-thrust ephemeris model.6 The reverse problem of escape from low-altitude circular orbits was investigated by Villac and Scheeres for multiple Jovian and Saturnian moons
through the use of Poincare maps.7
Alternatives to orbits deep in the gravity well of moons have been proposed as descope options to reduce propulsive cost. Russell and Brinckerhoff proposed the use of circulating eccentric
orbits that provide near-global coverage while avoiding low-altitude capture.8 Anderson et al.
avoid capture altogether by using heteroclinic and homoclinic connections between unstable peri-
*
Graduate Student, Aerospace and Engineering Mechanics, The University of Texas at Austin,
[email protected].
†
Assistant Professor, Aerospace and Engineering Mechanics, The University of Texas at Austin, 1 University Station,
C0600 Austin, TX 78712-0235, [email protected].
odic orbits to access varying viewing geometries and close approaches.9 More general transfers
between periodic orbits at different energies were researched by Davis et al.10
All the methods discussed seek new approaches for reducing the
required to meet science
objectives. In essence, all the
-saving techniques rely on exploiting third-body perturbations.
These third-body perturbations make it difficult to readily compare the different mission architectures proposed. The
cost of heavily perturbed capture trajectories are highly dependent on
specific boundary conditions and are expensive to compute and optimize. In this work, a simple
analytic expression is derived for the minimum
required to transfer between the bounding energy levels associated with capture. This floor on
provides mission designers with a valuable
bound for low energy captures to a planetary moon orbit. The analytical minimum
is numerically validated with a variety of optimized third body transfers. The results are applied across a
spectrum of energies for several other planetary moon bodies of interest.
MODEL
Circular Restricted Three-Body Problem (CRTBP)
Figure 1.The CRTBP rotating frame and coordinate system.
The framework used for the dynamics is the barycentric form of the circular restricted threebody system.11 The frame rotates about the barycenter of the system. Figure 1 depicts the plane of the rotating frame and coordinate system used in the CRTBP. The system is normalized
using the mass ratio of the smaller primary to the combined mass of the system, defined as . In
addition, derived length and time units (LU and TU) are used where LU is the mean distance between the two primaries and TU is set such that the system angular velocity is one radian per time
unit.
The normalized equations of motion for the third body in the CRTBP are described by Equation (1). The numbered subscripts 1 and 2 relate to the primary and secondary respectively:
(1)
where
is the spacecraft position,
is velocity,
ized frame rotation rate, and is the pseudo-potential:
2
is the normal-
with the distances from the respective bodies. There is one integral of motion known as the Jacobi constant, defined by Equation (2) when using the selected frame and normalized units.
–
where
is the total velocity,
(2)
.
System Definition
Jupiter and Europa are selected for in-depth analysis due to significant scientific interest in the
planetary moon. The Decadal Survey named Europa as the top outer planet destination, leading to
numerous mission design studies.12 In particular, the Europa Clipper is an ongoing mission concept under study by NASA’s Jet Propulsion Laboratory to perform science at Europa via multiple
flybys.13 Table 1 gives the Europa system parameters used in the CRTBP model.14
Table 1. Jupiter-Europa Three-body System Parameters
Parameter
Value
Europa Semi-Major Axis (km)
6.711 105
Europa Mean Radius (km)
1560.70
Jupiter Mean Radius (km)
71492.0
Europa GM (km3/s2)
3202.73879
Jupiter GM (km3/s2)
1.26686535 108
2.5280175 10-5
48,843.88
(s)
THEORETICAL MINIMUM TRANSFERS
For a given transfer between two orbits it is expected that there is a lower limit on
for impulsive maneuvers. It is assumed that such a transfer departs the initial orbit for free, as in the
case of unstable manifolds. Further assume that this theoretical trajectory intersects the desired
final orbit at a suitable location such that a single tangential maneuver is required for capture.
These assumptions result in a single-maneuver trajectory that is expected to have the most efficient burns possible. This single-maneuver is performed solely to change the Jacobi constant of
the orbit. The result is a boundary value problem of finding the minimum
required to obtain a
desired . Taking the derivative of Equation (2) gives that
, as only the velocity can
be changed. It is therefore apparent that higher velocities will require less
to obtain the same
change in Jacobi integral.
Finding the minimum
becomes reduced to finding the location of maximum velocity. To
reduce the scope of the problem the capture burn is limited to a given radius from the smaller
primary, making a constant and reducing the problem to two variables. Spherical coordinates
of latitude ( ) and longitude ( ) centered at the smaller primary are introduced to represent the
capture location as shown in Equation (3).
(3)
For synchronous tidally locked moons, these coordinates are in agreement with IAU conventions. New terminology is introduced to describe general locations on the sphere. “Interior” and
“exterior” are used to describe the sides closest to and furthest away from the primary. Leading
3
and trailing denote sides of positive and negative respectively, based on the revolution of the
secondary. Poles refer to the top and bottom of the sphere.
The minimum
expression is initially developed in the CRTBP, where the transfer has
boundary value conditions in the form of the initial and final Jacobi constants. The expression for
Jacobi integral is rearranged to solve for total velocity, , and expressed in terms of spherical coordinates in Equation (4).
–
(4)
Derivatives with respect to latitude and longitude are taken to find locations of extrema, given
by Equation (5):
(5a)
(5b)
The first property to note is that the derivatives are independent of Jacobi constant. The locations
of extrema are valid regardless of the integral of motion, although extrema locations can be inaccessible if
is negative. Looking at the latitude derivative in Equation (5a) there are three cases
leading to an extrema. First it is seen that zeros will exist at the equatorial latitude due to the leading sine term. The bracketed term will be zero at the poles technically under specific conditions
for longitude, but for practical purposes the poles are invariant to longitude. More complicated
zeros exist when the bracketed term is zero over a range of other latitudes and longitudes that do
not have an obvious analytical expression. The longitude derivative shown in Equation (5b) has
three cases of zeros: at the poles, the interior and exterior longitudes (
, and everywhere
that is unitary. The first two cases remain constant but it is clear that the last location will shift
as varies as is a function of . A summary of the possible conditions for each derivative is
given below, where represents the full bracketed term in Equation (5a):
Assessing pairs of latitude and longitude derivatives gives the locations of the extrema. For
practical purposes all pairs involving the poles are considered repeating. Accounting for trivially
infeasible or repeating pairs gives a total of ten extrema locations. The most obvious extrema are
at the poles of the moon as both derivatives are zero-valued at these locations for multiple pairs.
A second clear set is the intersections of the equatorial latitude with the interior and exterior meridians (
. Two more are found along the equator when
is unity. Solving this
pair for the longitudes gives
. The final two unique extrema locations arise from
the intersection of
and the meridians. An analytical expression for the latitudes is:
(6)
Note that the cosine of latitude will always be positive, resulting in a negative second term. In
order for a solution to exist, the multiplication of the first two terms must be positive. However,
4
when the longitude is
it is found that is always greater than unity resulting in a negative
value. This result indicates that only the interior meridian longitude of is an extrema. Substituting in for the analytical expression for the latitudes then becomes:
(7)
Equation (7) is transcendental and requires a numerical solution. A basic root-solver is sufficient
to solve for the latitude. Since there are known bounds the bisection method is utilized to find the
latitude between
and
and the second latitude is the negative of that value. Note that the
elimination of the exterior meridian locations from the same-sign terms in Equation (6) reduces
the total number of extrema to eight unique locations.
The Hessian and its eigenvalues are computed to characterize each extrema. All positive eigenvalues indicate the matrix is positive definite and the extrema are minima. The opposite is
valid if there are all negative eigenvalues. A combination of positive and negative eigenvalues
results in a saddle point.
(8a)
(8b)
(8c)
(8d)
It is found that intersections of the meridians and the equator are velocity maxima. The minima occur along the interior meridian when the bracket term is zero. All other extrema are saddle points. Extrema are denoted as
and grouped into similar pairs, such as the poles or the
equatorial-meridian intersections.
Table 2. Summary of velocity extrema.
Extrema
Conditions
Type
Saddle
Maxima
Minima
Saddle
5
Figure 2. Extrema locations at Europa for a capture radius of 5000 km with contours of
(km2/s2)
at
and extrema locations
(star)
(diamond)
(square) and
(triangle).
Table 2 gives a summary of the extrema locations as well as the type of extrema. The derived
equations are used to solve for the extrema locations at Europa using a sphere radius of
km. This altitude is chosen to exaggerate the deviations of
and
from the poles and
leading/trailing edge respectively. Even at this capture radius the minima points
and
remain
close to the poles at latitudes of
. The saddle points
and
similarly have only a
slight offset from the leading and trailing sides of Europa, with longitudes of
and
. Figure 2 details the extrema locations, including contours of
for
to verify
the extrema type.
From the symmetry of the equations in and , each pair of minima share equal
and therefore are global minimums. Of the two maxima, the one with the larger
is the global maximum.
Calling the squared velocity of the maxima at the exterior
and at the interior
and taking
the difference gives Equation (9):
(9)
Evaluating the sign of this expression for varying radii indicates that the interior velocity is greater while the capture radius is less than 1 LU. This limit is well beyond any reasonable capture
value and the interior velocity is therefore considered to always be the global maximum.
With knowledge of the extrema it is possible to assess the optimal locations for doing burn
maneuvers. The actual
can readily be found by solving for the velocities for two given values
of using Equation (4) and taking the difference. Substituting in for the location of the velocity
maximum gives Equations (10) and (11):
–
(10)
(11)
A practical example of calculating the minimum
is as follows. The starting orbit is a 1:1
resonance orbit with an initial Jacobi constant of
. The only condition for the
final orbit is that it is captured about Europa with no possibility of departure. The minimum
change of the Jacobi integral that satisfies this requirement occurs when the Hill's throats close
6
and the zero velocity surface prohibits escape. Numerically, this transition is found to occur at
. Two additional captures with
and
are additionally
considered to allow comparisons with captures at larger
. These boundary conditions are used
to calculate the expected minimum
using Equation (10) and Equation (11) over the entire feasible range of capture radii. The range is from Europa’s radius
) to the location of the Hill’s
throat at L1 (8.69 ). There is also interest in calculating the worst case scenario,
, as a
comparison point. Equation (4) is used to get the minimum velocities
and
at the near-polar
locations. As these locations require root-solving for latitude an analytical expression is not derived.
of
Figure 3. (Left) The maximum and minimum (dash)
to capture about Europa for three values
. Markers are left off of
lines for clarity. (Right) The difference between the maximum
and minimum
.
The maximum and minimum
s are plotted in Figure 3 along with their difference for comparison. The curves end at maximum
as the maneuver locations become inaccessible for the
given final Jacobi constants. Because velocity increases as radius is decreased, the most efficient
maneuvers occur just above the surface of Europa for all cases. The difference between the two
maneuvers increases near-exponentially with capture radius. The largest change occurs for
with the difference ranging from 0.27 m/s to 237 m/s where the maximum
curve
ends. Higher values of
have similar differences, but occur over a decreased span of capture
radii. Assuming the capture maneuver can occur at low altitudes of the moon it is therefore not
critical to aim for the minimum
location. The difference between best and worst case scenarios at low altitudes (such as 200 km) are typically less than 1% of the total cost. Where mission
requirements necessitate a higher altitude capture, targeting the minimum
can provide savings
on the order of 100 m/s or more.
The theory can be expanded to calculating the minimum
over a range of boundary conditions. Appropriate selection of the range allows comparisons between potential orbit options. Two
capture options are considered. First is a 200 km, 95 inclination circular orbit representative of
typical science orbit designs.15,16 The remaining two-body orbital elements and
, are considered free variables. This freedom implies that the orbit Jacobi constant will vary based on the
specific capture. Calculating the range of requires considering all possible captures. Pairs of
are gridded over with each pair and the inertial state relative to Europa is calculated.
7
The inertial state is converted to the rotating CRTBP frame and the Jacobi constant is calculated.
A range of
is found for the low-altitude science orbit.
The second capture options is a high-altitude loosely captured state. Work by Russell revealed
an approximate limiting case of maximum altitude of 7000 km for stable orbits at Europa with
inclinations near
. The range of Jacobi integral for this family is given as
.17 To consider a broad range of possible arrival conditions the initial values used are
from a relatively poor arrival of
up to a perfect arrival of
. This range includes the L2 halo orbit family which is between 3.0008370 and 3.0033259 at Europa.
Figure 4. Minimum
(km/s) to capture into low altitude (left) and high altitude (right) orbits
for varying boundary conditions. Dash lines indicate the range for L2 halo orbits.
Gridding over the boundary values for both capture conditions results in the contours shown in
Figure 4. Dashed lines are included to show the bounds of the L2 halo orbit family. Starting with
the low-altitude capture, the range of is small resulting in near-invariant
as the end boundary value is varied. It can be seen that transfers starting at within the L2 halo family have expected minimum
s from approximately 400 m/s to 500 m/s. Larger differences in the boundary
conditions result in maneuvers up to 1 km/s for the considered range, although this maximum
represents a poor arrival uncommon for typical mission design.
The values of
for the 7000 km altitude capture occur over a broader range resulting in notable variation of
. The lowest burns can be seen with the minimum value of
as it is closest
to the initial boundary values. Transfers from the L2 halo orbits at this
have expected
s between 30 m/s and 350 m/s. The 30 m/s case occurs due to the largest
from the halo orbit family having a relatively small
from the minimum . As
is increased the
from the halo orbits increases to a maximum of 600 m/s. This result indicates that simply using a
high-altitude capture does not guarantee
savings from low-altitude capture. Recall from Figure 3 that higher altitude captures lead to greater propulsive costs when
remains constant. Reducing
therefore requires sufficient decrease in
compared to low altitude orbit captures.
With this consideration, appropriate selection of the boundary conditions can lead to
savings
on the order of hundreds of meters per second.
8
Hill’s Approximation
While the contour of minimum
can be rapidly generated the process must be repeated between specific three-body systems. In order to generalize the analysis to any three-body system
the expressions are re-derived using the Hill model. This formulation is the limiting case of the
CRTBP as goes to zero, using a frame centered at the secondary.11 By maintaining the use of
normalized dimensionless units this allows results to be scaled to any system. Normalization requires the use of a different length unit such that
, where
angular velocity of the smaller primary. The time unit remains as
velopment of this model to the work is a change in the motion integral:
is the dimensional
. The pertinent de(12)
Table 3. Summary of parameters for planetary moons, and the predicted minimum
range to capture from the L2 halo orbit family into low and high altitude orbits
Secondary
Moon
Phobos
Deimos
Io
Europa
Ganymede
Callisto
Amalthea
Himalia
Elara
Pasiphae
Sinope
Lysithea
Carme
Ananke
Leda
Thebe
Adrastea
Metis
Mimas
Enceledaus
Tethys
Dione
Rhea
Titan
Hyperion
Lapetus
Phoebe
Ariel
Umbriel
Titania
Oberon
Miranda
Triton
Nereid
Charon
Radius (km)
1,738
11
6
1,815
1,561
2,631
2,400
98
93
38
25
18
81
20
15
8
50
10
20
196
250
530
560
765
2,575
148
730
110
579
586
790
762
240
1,353
170
593
(km)
88,454.7
23.1
33.8
15,196.3
19,696.11
45,733.4
72,283.6
282.7
19,592.3
8,645.0
10,897.8
8,196.2
4,008.2
8,308.9
5,756.7
1,600.0
163.4
27.7
47.0
798.8
1,205.2
3,215.0
4,632.3
8,636.0
75,714.8
4,601.2
53,010.4
11,501.6
4,766.7
6,328.1
14,982.5
19,014.2
1,204.4
21,075.1
32,280.8
11,931.7
(hr)
104.36
1.22
4.82
6.76
13.57
27.33
63.75
1.90
957.09
991.80
2,807.49
2,895.35
990.15
2,643.25
2,410.24
911.84
2.58
1.14
1.13
3.60
5.23
7.21
10.45
17.26
60.91
81.27
303.02
2,102.68
9.63
15.83
33.25
51.43
5.40
22.45
1,375.62
24.40
(km)
200
----200
200
200
200
--11.9
5.00
5.00
5.00
10.4
5.00
5.00
5.00
--------32.0
67.9
71.2
98.0
1000
19.0
93.5
14.1
74.2
75.1
101.2
97.6
30.6
173
21.8
76.0
609 – 645
----353 – 617
377 – 512
602 – 727
601 – 664
--31.6 – 32.0
14.0 – 14.2
8.43 – 8.49
6.12 – 6.16
2.85 – 3.03
6.53 – 6.58
4.62 – 4.66
2.18 – 2.22
--------2.01 – 42.6
31.4 – 97.4
66.6 – 120
115 – 164
548 – 628
29.8 – 33.0
153 – 159
5.83 – 6.00
73.5 – 133
87.2 – 128
167 – 200
168 – 191
4.33 – 42.4
296 – 373
34.3 – 34.8
190 - 225
(km)
29,698
-2.79
6.01
3,586
5,439
13,623
23,290
2.47
6,870
3,034
3,848
2,895
1,344
2,933
2,031
561
8.07
-0.16
-3.30
87.9
178
613
1,086
2,304
24,334
1,487
18,110
3,978
1,115
1,663
4,535
5,996
188
6,137
11,303
3,648
(m/s)
18.8 – 341
--0.16 – 2.82
50 – 905
32.2 – 585
37.2 – 674
25.2 – 457
--0.45 – 8.26
0.19 – 3.51
0.08 – 1.56
0.06 – 1.14
0.09 – 1.63
0.07 – 1.27
0.05 – 0.96
0.04 – 0.70
1.41 – 25.5
----4.93 – 89.4
5.12 – 92.8
9.91 – 180
9.85 – 179
11.1 – 202
27.6 – 501
1.26 – 22.8
3.89 – 70.5
0.12 – 2.20
11.0 – 199
8.88 – 161
10.0 – 181
8.22 – 149
4.96 – 89.8
20.9 – 378
0.52 – 9.45
10.9 – 197
As the dynamics of the three-body system are preserved near the secondary in the Hills model,
the locations of the maximum velocities derived in the previous section are unchanged. Solving
for the velocity gives:
(13)
9
Evaluating Equation (13) at the location for maximum velocity for the boundary values of the
integral:
(14)
Taking the difference of the velocities at both boundary conditions from Equation (14) gives
the theoretical minimum
as a function of the capture radius. As discussed, by using dimensionless units this
can be scaled to any three-body system. Given the system parameters the
normalized length and time units, LU and TU, are calculated. Basic unit conversion can then be
applied to the
to scale results to kilometers per second. A summary of the scaling parameters
for a majority of planetary moons is given in Table 3.18
A grid approach similar to the CRTBP results from the previous section is used to find
over a range of capture conditions at Europa. Converting to normalized units, the low altitude
capture radius is at
, while the high altitude capture radius is at
. The
range of motion integrals for the low altitude becomes
and for
the high altitude
. The halo orbit family has
.
Figure 5. Normalized
in the Hill model for capture at Europa with
(right). Dash lines bound L2 halo orbit
(left) and
The
contours for the captures at Europa are given in Figure 5 using normalized units, with
contours similar to those seen in Figure 4. For the low altitude capture the velocity range for halo
orbit transfers is
, while the high altitude orbit is
.
As discussed, a key benefit of using the Hill’s formulation is scalability to other three-body
systems. This scaling property can be applied to the high altitude orbit results as the capture orbit
is defined as the stability limiting orbit, which is invariant between systems in the Hill model.
Note however that the altitude of the orbit, , will vary between bodies. The results are scaled to
the planetary moons, resulting in the maneuver magnitudes listed in Table 3. For very small
moons such as Phobos the altitude is actually below the mean body radius and so no
results
are given, although the bodies are still listed for completeness. It can be seen that Europa’s values
closely match those found from the CRTBP results for verification between the two models.
Low altitude orbits are defined relative to the body, making them system specific. Due to this
variation the capture radius as well as the motion integral range for the capture orbits must be
recalculated for each body. For the purposes of this work, low altitude (
is defined as follows:
10
km, where
is the radius of the planetary moon. An exception
to this case is Titan, which uses an altitude of 1000 km due to its thick atmosphere. Additionally,
an absolute minimum feasible altitude of 5 km is enforced in consideration of the highly irregular
shapes and gravity fields of smaller moons. The body-specific altitude is simply added to the
body radius to get the capture radius . Orbits with resulting capture radii greater than
are considered to be effectively the same as the high altitude case and no
is calculated.
As with the Europa example, the two-body orbital elements are used to get initial states for each
moon, leading to a range of values for the low altitude orbits.
The results of the minimum
are given in Table 3 for most planetary moons of interest.
Several of the smallest bodies (Phobos, Metis, etc.) are sufficiently small that no feasible captures
exist for the given capture options as the orbit altitudes are too close or even below the surface of
the moon. This is not to claim that capture is impossible; alternative candidates, such as threebody retrograde orbits, exist and could be used in a later analysis. Other moons, such as Deimos,
have “high” altitude captures where the altitude is near to or less than the low altitude definition.
This result indicates that the moon is too small for an accurate two-body approximation, so no
low altitude orbit result is given.
Comparing the
s at Europa shows reasonable agreement between the three-body and Hill’s
model approaches, verifying the accuracy of scaling results. A majority of the large moons of interest, such as the Galilean moons or Titan, all require low altitude captures on the order of hundreds of meters per second. As expected, smaller moons cost less, down to only a few meters per
second. Considering the high altitude captures shows the potential for significant cost savings as
large as two orders of magnitude, assuming the initial halo orbit is sufficiently close to the capture orbit.
THEORY VALIDATION
To validate the analytical minimum
results multiple physical transfers are found and the
total
s are compared with the predicted minimums. For an initial approach the transfer is selected to be from orbits in the L2 family at Europa to a moderately constrained low-altitude science orbit. Use of the L2 orbits reduces the scope of possible departure times to one period of the
orbit. Capturing into a low-altitude science orbit ensures that sub-optimal transfers will be significantly more expensive due to the need for non-tangential burns such as inclination changes. Starting at an initial L2 halo orbit, transfers are found using a grid search method with a differential
corrector to improve grid points. The periodic orbit Jacobi constant is then varied to a new halo
orbit. As the grid search is time intensive continuation methods are utilized to find transfers for
the new orbit, using the transfers of the previous orbit as initial guesses.
11
L2 Halo Orbits
Figure 6. Orbits from the Halo family about the L2 point, axis centered on Europa.
The existence and initial conditions of L2 halo orbits is well documented in the literature. Precomputed orbits from Reference 17 are used to generate the family. Example orbits of this family
are given in Figure 6, including the initiating planar Lyapunov orbit. The Jacobi constant for this
family range is
, spanning the critical value of a barely-closed
Hill’s throat.
Science Orbit Definition
Table 4. Summary of science orbit parameters.
Parameter
Value(s)
Semi-major axis, Inclination
1760.7 km, 95°
CRTBP X0
{1.0025983, 0.00, 0.00, 0.00, -0.0111789, 0.0977879}
Period
0.167933 (2.28 hr)
C
3.0095034
The science orbit is chosen to be characteristic of typical science orbits, namely high inclination, low altitude, and circular.16 These requirements are met with a circular, 200 km altitude science orbit at 95° inclination. For the scope of the current research the phasing of the science orbit
is ignored and the phasing orbital elements are set to zero. As the orbit is calculated in the threebody frame the actual orbital elements are varying. A summary of the science orbit’s relevant
parameters is given in Table 4.
Ballistic Transfer Grid Search
A global grid search method is employed to find transfers between the two boundary value orbits. It is assumed that low
transfers are characterized by two tangent-only burns. The first
burn is applied to depart the gateway orbit onto some transfer trajectory. The transfer trajectory
then ends with a science orbit insertion burn. It is further assumed that the science orbit insertion
burn occurs when the distance from the smaller primary is at an extrema, similar to apses in the
two-body problem. These assumptions reduce the problem to two variables. The first is the gateway orbit location of departure, , defined to be the normalized arclength distance along the
gateway orbit. This definition gives an initial value of
with a final value of
. The
12
parameter is calculated as a function of time by integrating Equation (15) along with the CRTBP
equations of motion to get the total distance , then normalizing. The state becomes a function of
this new parameter,
.
(15)
The second grid variable is the magnitude of the departure burn,
. Positive values represent
a burn in the direction of the velocity. Unlike this variable is not naturally bounded. Setting
m/s is considered generally sufficient for feasible transfers. Recall that the minimum
development assumed a negligible maneuver at the initial orbit. It is therefore expected for the
best solutions to have low initial . In some cases, these initial burns are trivially small, and represent catalysts to precipitate a transfer onto an unstable manifold. The two variables, and
,
are gridded over their respective ranges to create multiple pairs of departure conditions. Each pair
leads to an initial state that is integrated forward by a maximum allowed flight time. The resulting
trajectory is checked for locations where the distance from the smaller primary is at an extrema
using the dot product of position and velocity. The apse with an altitude closest to the science
orbit is chosen as the location for the orbit insertion burn. At this location the osculating orbital
elements are calculated to compute the
required to capture into a circular orbit. As stated this
burn is tangential and does not match the desired orbit inclination.
Repeating this process over every pair of (
) results in global knowledge of the two-burn
transfer trajectories and their resultant capture orbital elements. The inclination and semi-major
axis are combined into a square difference objective function to assess how well grid pairs approach the target values. The function is given by Equation (16), where
is the difference
between the actual and desired orbital elements. Scaling factors are used so that inclination and
semi-major axis are equally weighted.
(16)
The grid values of are used to identify regions where the transfers nearly approach the desired science orbit, however it is unlikely that any given grid point is exactly zero. Gridpoints are
sorted according to their values and the best points are chosen for further improvement. The
points are improved with a simple differential corrector to minimize the objective function.
Table 5. Summary of initiating halo orbit parameters.
Parameter
Value
{1.0175667,
0.00,
0.0222484,
0.00, -0.0306484, 0.00}
(LU,LU/TU)
2.80006
(TU)
2
3.0011903
(LU/TU)
A summary of the halo orbit used in the initiating grid search is given in Table 5. The only
consideration for selecting this orbit is that it be sufficiently out of plane such that highly inclined
close approaches at Europa are feasible. The range of the burn magnitude variable
used is
m/s. The range favors negative burns as they are more likely to result in Europa close approaches. The grid step sizes are chosen with consideration of resolution versus grid
computation time. A balance of these considerations leads to selection of
and steps of
m/s for
resulting in 200 points of each grid variable for a total 40,000 grid points. Once
all grid points have been evaluated the lowest
points are selected for further improve-
13
ment. A basic Newton corrector is derived from derivatives of the semi-major axis and inclination
with respect to the transfer parameters:
(17)
where the derivatives are calculated using the finite difference method and the star superscript
indicates the targeted values. Solutions are considered feasible if the semi-major axis and inclination constraints are met to 100 m and 0.1 respectively.
Table 6. Summary of solution grid variables, state before exit burn and after capture burn,
target orbital elements, and burn magnitudes.
Parameter
Value(s)
{0.9454,-33.37 m/s}
{1760.700 km, 6.631e-13, 95.0000 }
485.98 m/s
519.34 m/s
Figure 7. The halo orbit (dash), transfer orbit, and resulting science orbit about Europa in the
CRTBP frame.
14
Figure 8. Transfer trajectory and capture in inertial space about Europa.
The
optimal solution that meets the science orbit within tolerance is occurs for
m/s. The total velocity change is 519.34 m/s which is dominated by the
science orbit capture burn. Total transfer flight time from burn to burn is 53.75 hours. A summary
of the solution parameters is given in Table 6, including the initial state after the exit burn and the
initial state after capture. The resulting transfer and science orbit are given in the CRTBP rotating
frame Figure 7 and in the inertial frame centered at Europa in Figure 8. The small magnitude of
the burn results in the transfer initially following the halo orbit before being attracted to Europa.
Figure 8 shows how the three-body dynamics twist the transfer trajectory such that the spacecraft
arrives at Europa with the desired target inclination.
It is possible to extend the grid search approach to non-periodic orbits by limiting the range of
departure times so that the associated can be calculated. Additional advancements can be made
to include science orbit phasing. Near-optimal transfers can then be used in state-of-the art optimizers as initial guesses for ballistic orbital transfers.
Transfer Orbit Continuation
As stated in the previous section, continuation methods are used to extend already found transfers as a function of . The halo orbit Jacobi constant is changed by a small perturbation and the
periodic orbit from the fully generated family is found. Following one transfer at a time, the differential correctors from the grid search are used to find transfers for the new periodic orbit. After
transfers have been found for the second halo orbit, linear extrapolation is used as a predictor step
to generate initial guesses for the succeeding periodic orbits. If necessary, the full grid search
method is repeated in an effort to find additional transfers that may not have existed for the original gateway orbit.
To aid the speed of the continuation a variable step size of
is used based on how well the
linear extrapolation guesses approximate the actual converged values, using the relative difference:
(18)
Note that the prime denotes the extrapolation guess value and is some small tolerance. If Equation (18) is satisfied the step size is increased. Corrector convergence failures result in a decrease
in step size down to a minimum. Further convergence failure ends the continuation process.
15
Figure 9. Example of three trajectories within the same “family” show similar capture locations. Initiating halo orbits not included to improve clarity of transfer trajectories.
The process to fully continue a single transfer is time consuming, with the majority of time
spent in the differential corrector phase. For this study, 16 unique transfer “families” are identified. These “families” exist in the sense that there are clear and continuous values of and
as
functions of that do not intersect with other “families”. Figure 9 shows an example of three trajectories within the same family. All three can be seen to share a general shape of gradually approaching Europa with convergence to nearly the same capture location.
Figure 10. Departure
(left) and capture
(right) for all identified L2 Halo based transfers as
Jacobi constant of the halo orbit changes.
Looking at Figure 10 one can see the evolution of the transfer
s. The first plot shows the
exit burn while the right shows the capture burn. For the exit burn case it is seen that there is no
general trend that can be applied to all families. Several families start with burns on the order of
500 m/s but quickly drop to zero as increases, while others start near zero and increase or
change non-monotonically. This indicates that starting Jacobi constant has little correlation with
the exit burn in a general sense.
Conversely there is an apparent trend in the plot for the capture burns. As Jacobi constant decreases most families show a decreasing trend. This trend is consistent with the analytic minimum
, and results from the boundary condions being closer in energy. Also of note is the limited
16
range in the maximum and minimum values of the capture
. All the families lie within 420520 m/s and most families vary by no more than 50 m/s through the entire range of . As expected, the total
optimal transfers have effectively no burn and resemble unstable manifolds
of the gateway orbit.
Figure 11. Total
for L2 Halo transfers remain above the predicted minimum (grayed area)
The theoretical minimum
are calculated over the range of halo orbit using the maximum and minimum motion integrals for the science orbits. The two lines of
are added to the
plot in Figure 11 and the area below the bottom line is grayed out. As the range of
for the science orbit is small there is negligible difference between minimum
s at constant halo orbit
Jacobin constant, resulting in an apparently single value. It is apparent from the figure that there
are no values below the theoretical minimum, although several may effectively be at the limit.
These near-minimum cases are transfers that closely resemble the assumed optimal conditions
made to derive the minimum
expressions. As discussed, the use of the unstable L2 halo orbit
family allows effectively free departure on unstable manifolds, some of which intersect the desired science orbit phase space. In general it is not expected for transfers to be near the ideal case.
Inclusion of science orbit phasing further limits allowed capture geometries and reduces the number of transfers that tangentially approach the target orbit.
CONCLUSIONS
The development of analytical expressions using the Jacobi constant to predict minimum
given two boundary conditions leads to a simple and effective tool for preliminary mission design. As the results do not require the calculation of physical transfers it is possible to rapidly assess multiple scenarios by simply changing the three design parameters. Any conceivable range
of boundary orbits, including science orbits and periodic orbits, can be evaluated so long as the
dynamics can accurately be modeled with the circular restricted three-body problem. By extending the expressions to the Hill model it becomes possible to perform one evaluation over a range
of design parameters and scale the results to any bodies of interest. Applications to a low-altitude
versus a descope high altitude orbit show that the descope option saves on the order of 100 m/s
when the capture burn occurs at low altitude.
While used primarily as a means to validate the minimum
theory, the grid search approach
developed to calculate transfers between periodic and science orbits has inherent utility for generating initial guesses of ballistic transfers. Multiple transfer families connecting the L2 halo family
at Europa to a low altitude science orbit showed verified that the minimum
expressions. The
17
optimal transfers closely followed the ideal transfer assumptions of a manifold departure with a
single, tangential capture burn.
ACKNOWLEDGMENT
This research was funded through the NASA Graduate Student Research Program (GSRP) and
conducted in part at the Jet Propulsion Laboratory with advisement from Anastassios Petropoulos.
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