AAS 05-383

SHOOT THE MOON 3D

Jeffrey S. Parker1 and Martin W. Lo2

Ever since 1990, with the launch of the Japanese Hiten Mission,1,2 the astro-
dynamics community has been fascinated by ballistic lunar capture trajectories.
An earlier work by Koon et al.3 provided an explanation of the dynamics and an
algorithm for designing such trajectories using invariant manifolds of periodic
orbits in the Planar Circular Restricted Three Body Problem (PCRTBP).
However, this algorithm took specific advantage of the topology of the planar
problem and could not be easily extended into the full spatial problem. The 2-
dimensional invariant manifolds of orbits in the PCRTBP have well-defined
interior and exterior regions within their 3-dimensional energy surface.
Conversely, halo and Lissajous orbits in the spatial Circular Restricted Three
Body Problem (CRTBP) have invariant manifolds of dimensions 2 and 3,
respectively, in a 5-dimensional energy surface. Topologically, such manifolds
do not have well-defined interior or exterior regions. The algorithms developed
to construct planar ballistic lunar transfers could not therefore be easily extended
to the spatial CRTBP. This paper addresses the Hiten Problem in the spatial
CRTBP case by using three-dimensional libration orbits as staging orbits both at
the Sun-Earth and the Earth-Moon Lagrange points to construct a ballistic
transfer from a low-Earth orbit (LEO) to a temporary capture orbit around the
Moon. The transfer requires no deterministic maneuvers other than the injection
maneuver from LEO. It requires a total ΔV of approximately 3.26 km/s from a
200-km circular LEO parking orbit, compared with an approximate 3.7 – 4.0
km/s for a conventional Hohmann transfer.

INTRODUCTION

The conventional approach to transfer from the Earth to the Moon may be approximated by a
Hohmann Transfer. A typical transfer from a 200-km low-Earth orbit (LEO) to a 100-km lunar orbit
requires approximately 3.91 km/s. This includes 3.05 km/s for the first maneuver and 0.86 km/s for the
second maneuver with a transfer time of approximately 5 days. If a mission merely requires the spacecraft
to be captured by the Moon, e.g., in a very high orbit, then the cost is reduced to approximately 3.7 – 3.8
km/s4.
Many efforts have been made to reduce the cost of a lunar transfer. In 1968, Charles Conley was one
of the first people to construct a low-energy lunar transfer based on the dynamics of the three-body
problem5. However, since the transfer time required several months, it was unsuitable for human missions
to the Moon and it was almost forgotten. Conley’s foundational work on the three-body problem is
fundamental to the methods described in this paper. In 1990, the Japanese Hiten mission (see Uesugi et al. 1
and Belbruno & Miller2) implemented a low-energy transfer trajectory to the Moon using Weak Stability
Boundary Theory which allowed the spacecraft to become temporarily captured by the Moon using less
energy than that required by a standard Hohmann transfer. In honor of the Hiten mission, this paper refers
to the ballistic transfer and lunar capture problem as the “Hiten Problem”. The community quickly realized

1
Doctoral Student, Colorado Center for Astrodynamics Research, University of Colorado at Boulder, 431 UCB, Boulder, CO 80309-
0431. E-mail: parkerjs@colorado.edu.
2
Visiting Associate, Computer Science, California Institute of Technology. E-mail: Martin.Lo@jpl.nasa.gov.
that invariant manifolds play a key role in the Hiten Problem; however, there were no explicit numerical
algorithms or computations showing how this can be done using invariant manifolds. In 2000, Koon et al.3
provided an algorithm that used numerical computations to show how invariant manifolds can be used to
reproduce a Hiten-like mission. For clarity and simplicity, their study used invariant manifold theory in the
planar bicircular model3 to prototype a trajectory that transferred from LEO to a ballistic lunar capture
orbit. The resulting trajectory required only one small 34 m/s maneuver, which may be eliminated with
optimization, to become captured by the Moon instead of the large conventional maneuver required by the
Hohmann transfer. This paper refers to ballistic lunar transfer trajectories as “Shoot the Moon” trajectories
in reference to the work of Koon et al.3
The present study extends Koon et al.’s work by constructing three-dimensional ballistic lunar
transfers using dynamical systems theory. Three-dimensional trajectories have many advantages, including
the following features: they may be designed to efficiently implement most inclined LEO parking orbits,
they provide better geometry for communication purposes, and they provide access to any final lunar orbit,
including polar orbits. Transfer trajectories may also be designed to rendezvous with other spacecraft or
space stations in other three-body orbits, such as halo orbits about the lunar L 1 point. Many of these
transfers may be designed to require much less energy than conventional transfers. Ultimately, this allows
for larger payloads or smaller rockets.

This paper is organized in the following manner. Section 2 presents the most important aspects of
the Circular Restricted Three Body Problem (CRTBP) for the Hiten Problem, including the three-
dimensional families of halo orbits found about the first and second Lagrange points. Section 3 then
introduces the Sun-Earth-Moon-Spacecraft four-body problem and discusses a means to decouple the four-
body problem into two three-body problems. Section 4 summarizes the results of the work of Koon et al. 3
to construct planar lunar transfers. Section 5 extends the lunar transfers into three-dimensions and verifies
the transfers using the bicircular model. Finally, Section 6 discusses the conclusions and future directions
of this work.

THREE-BODY PROBLEM

The spatial CRTBP is used in this study to model the motion of a spacecraft in the Earth-Moon
system and in the Sun-Earth system. This section introduces the important aspects of the CRTBP for the
Hiten Problem, including some well-known three-dimensional orbit solutions. For more information about
the CRTBP, see Szebehely6 for instance.

Definition of the CRTBP

The CRTBP places two primary bodies, e.g., the Sun and Earth-Moon barycenter or the Earth and
Moon, in circular orbits about their barycenter and uses their gravitational attraction to approximate the
motion of a massless third body. The primary bodies are assumed to be point-masses and no other forces
or perturbations are included in the model. In this paper, the system that is referred to as the “Sun-Earth
system” is shorthand for the three-body system where the larger primary mass is the Sun and the smaller
primary mass is the Earth-Moon barycenter.
The coordinate system of the CRTBP is centered at the barycenter of the primaries and set to rotate
with the motion of the primaries about the origin; that is, the synodic frame rotates in a counter-clockwise

3
The bicircular model for the four body Sun-Earth-Moon-Spacecraft system models the orbits of the Earth
and the Moon as coplanar circles around the Sun and Earth, respectively, and the spacecraft as an
infinitesimal point mass. See Section 5, below.

2
fashion compared to the inertial frame, assuming the conventional right-hand rule for the angular
momentum in the system. The x-axis, also known as the syzygy axis 6, extends from the origin through the
smaller primary, the z-axis extends in the direction of the angular momentum of the system, and the y-axis
completes the right-handed coordinate frame. It is convenient to normalize the units in the system such that
the following metrics are equal to one: the distance between the two primaries, the sum of the mass of the
two primaries, and the gravitational parameter. The orbital period of the two primaries is then equal to 2π.
The three-body constant, μ, is defined as the ratio of the smaller primary’s mass (m2) to the sum of the mass
of the two primaries (m1+m2), approximately equal to 0.0121506 for the Earth-Moon system and
3.0404025 x 10-6 for the Sun-Earth system7. Since the system has been normalized, the coordinates of the
primaries in the rotating axes are therefore equal to [-μ, 0] and [1 – μ, 0] for the larger and smaller primary,
respectively. The mass of the larger and smaller primaries are equal to μ and 1 – μ normalized mass units,
respectively. The mass of the third body is neglected in the model.

The equations of motion for the third body in the rotating frame are equal to:6

x+µ x −1 + µ
x = 2 y + x − (1 − µ ) −µ
r13 r23
y y
y = −2 x + y − (1 − µ ) −µ (7)
r13 r23
z z
z = −(1 − µ ) −µ
r13 r23

where r1 and r2 are equal to the distance from the third body to the larger and smaller primary, respectively:

r12 = ( x + µ ) 2 + y 2 + z 2
(8)
r22 = ( x − 1 + µ ) 2 + y 2 + z 2

In this form it is clear that the dynamics of the system depend only on the mass fraction μ. Furthermore,
when μ goes to zero, the dynamics approach the two-body dynamics in a rotating frame.

It is well known that five equilibrium points exist in the CRTBP 6, referred to as the Lagrange points.
In this study we will adopt the convention that
L1 lies between the two primaries, L2 beyond 300 LL4
the smaller primary, L3 beyond the larger
200
primary, L4 above the x-axis, and L5 below the
x-axis. Figure 1 shows a plot of the locations of 100
y (x103 km)

LL3 Earth LL1 LL2

the Lagrange points in the Earth-Moon rotating 0
Moon
coordinate system, where the Lagrange points -100

have been labeled LL1 – LL5 for conventional -200

reasons to avoid confusion with the Earth’s LL5

-300
Lagrange points.
-400 -300 -200 -100 0 100 200 300 400 500
x (x103 km)
The five Lagrange points are the only
fixed-point solutions to the CRTBP. Many
periodic orbit solutions exist including planar Figure 1. The Earth, Moon, and five Lagrange points in
the rotating coordinate system of the CRTBP.

3
and three-dimensional libration orbits. For discussions of periodic orbits, see Broucke8 or Hénon9 – 12 for
instance. The three-dimensional libration orbits will be briefly discussed in the following section.

4
Libration Orbits
Libration orbits are two and three-dimensional periodic and quasiperiodic orbit solutions to the
CRTBP that exist about the three collinear Lagrange points. These orbits are usually unstable. A particle
on a three-dimensional libration orbit will oscillate about the corresponding Lagrange point with two
independent frequencies: an in-plane frequency and an out-of-plane frequency. If the frequencies are equal
then the particle is on a periodic orbit commonly known as a halo orbit 13. Otherwise, the particle is on a
quasiperiodic orbit, such as a Lissajous orbit.13, 14 The ISEE3 mission15 was the first mission to fly on a halo
orbit and many have flown since. A portion of the family of halo orbits about the Earth-Moon L 1 point is
shown in Figure 2.
4
x 10
Halo LL Orbits
1

Moon’s 0
Orbit
-2 L1 Moon
z (x104 km)

-4

-6

-8

5
0
4
x 10 -5 3.8 4 4.2
2.6 2.8 3 3.2 3.4 3.6
y (km) 5
x (km) x 10

Figure 2. A portion of the family of halo orbits about L1 in the Earth-Moon system.

Invariant Manifolds
The libration orbits shown in the previous section are typically unstable. If a particle is randomly
perturbed while on the orbit, the particle will fall off of the orbit at an exponential rate. In Hamiltonian
systems, not only are there asymptotic orbits departing from the unstable orbits, there are also asymptotic
orbits approaching the unstable orbits. The set of all trajectories that asymptotically depart from the
unstable orbit is known as the orbit’s unstable invariant manifold; the set of all trajectories that
asymptotically arrive onto the unstable orbit is likewise known as the orbit’s stable invariant manifold.
Figure 3 shows the stable and unstable invariant manifolds for a typical halo orbit about the Earth-Moon L2
point. The manifolds are very similar for orbits about the L1 point. One can see the underlying tubular
structure in the manifolds. As the manifolds approach one of the primaries, this structure begins to break
down due to the large divergent behavior near the primaries.

5
 Stable Invariant Manifold Unstable Invariant Manifold
120
60
100
40
80 Periodic Orbit
Moon’s Orbit
20
60
Motion 0
Motion
Y (x103 km)

Y (x103 km)
40
-20
20 Moon
-40
0
-60
-20 Moon
-80
-40
-100
-60 Moon’s Orbit
-120
300 350 400 450 500
Periodic 550
Orbit 600
300 350 400 450 500 550 600
X (x103 km)
X (x103 km)
Figure 3. The stable (left) and unstable (right) invariant manifolds for an orbit about L2.

Mission designers may use these invariant manifolds to model the motion of spacecraft in their
vicinity. If a mission’s objective is to transfer onto an unstable periodic orbit, then the spacecraft need only
target that orbit’s stable manifold in order to insert into that orbit. Although missions such as ISEE3 and
Hiten were not designed using invariant manifolds explicitly, the underlying dynamics may be understood
using invariant manifold theory. The advantage of the dynamical systems approach is the ability to
compute and visualize global families of low-energy transfer trajectories, giving mission designers a priori
knowledge of the underlying dynamics in the libration orbit regime.

FOUR-BODY PROBLEM

Spacecraft in the near-Earth environment are influenced substantially by the Sun and the Moon,
suggesting that a simple three-body analysis of the near-Earth environment is not sufficient to describe all
of the dynamics observed in the Earth’s neighborhood. A very interesting aspect of the Sun-Earth-Moon
system is that the invariant manifolds of orbits about the Earth’s L1 and L2 points (EL1 and EL2,
respectively) intersect the invariant manifolds of orbits about the Moon’s L1 and L2 points (LL1 and LL2,
respectively). Thus, objects may transfer between the Earth-Moon three-body system and the Sun-Earth
three-body system and escape or become captured by either system. This characteristic of the Sun-Earth-
Moon system will prove to be very useful when transferring from LEO to a lunar libration orbit.

Decoupling the Four-Body System

Since the four-body problem is more complex than the three-body problem, it helps to decouple the
Sun-Earth-Moon-Spacecraft four-body problem into two three-body problems. Then one can use standard
three-body analysis tools to determine the characteristics of solutions in different regimes of the four-body
problem. In this paper, a method has been used to decouple the four-body system that is very similar to the
method that mission designers use to approximate gravity-assisted trajectories, namely, the patched-conic
method that uses the notion of a body’s sphere of influence (SOI). For instance, if a particle traveling in
the Sun-Earth three-body system approaches the Moon, it may cross an invisible boundary after which the
Moon’s influence becomes stronger than the Sun’s. At that boundary, the motion of the particle is better
approximated by propagating it in the Earth-Moon three-body system rather than the Sun-Earth three-body
system. Once the particle departs from the Moon’s vicinity, it may then re-cross that three-body boundary
once again. In this paper, the three-body boundary is referred to as the Earth-Moon three-body sphere of
influence (3BSOI). Using an analytical approximation that will be presented in a future paper, the 3BSOI
may be approximated by a sphere centered at the Moon with a radius of about 160,000 km. This surface
includes both the lunar L1 and L2, but not the other three libration points; see Figure 4 for a plot of the
system.

6
 6
x 10
1

Earth’s Orbit

0.5 3BSOI LL5 Moon’s Orbit

To the Sun
0 EL1 Earth EL2

LL4
LL3
-0.5

-1
1.48 1.485 1.49 1.495 1.5 1.505 1.51
8
x 10
Figure 4. A plot of the Earth-Moon 3BSOI in the Sun-Earth reference frame.

2D SHOOT THE MOON TRANSFER

In 2000, Koon et al.3 used dynamical systems theory to reproduce the Hiten trajectory in two-
dimensions. Figure 5 shows two example trajectories of such transfers to the Moon in the Sun-Earth
rotating frame. o o
Transfer: h = 209.8 km, α = 234.105 , φ = 0.7986877 Transfer: h p = 175.1 km, α = 39.321 , φ = 0.4205889
p

0.6 1.5

0.4

0.2
Earth EL2 1
0EL1 Moon’s
-0.2 Orbit
Moon
y (x106 km)

y (x106 km)

-0.4 0.5 3BSOI

-0.6 Moon’s
-0.8
3BSOI Orbit Moon
0EL1 EL2
-1 Earth
-1.2

-1.4 -0.5

148.5 149 149.5 150 150.5 151 148.5 149 149.5 150 150.5 151
x (x10 6 km) x (x10 6 km)

Figure 5. Two examples of dynamical systems transfers to the Moon. Left: a trajectory that skims EL2 before
targeting the Moon; right: a trajectory that implements an EL1 encounter before targeting the Moon.

The total cost of the trajectory shown in Figure 5 is less than the cost of a typical Hohmann transfer,
but the trade-off is transfer time. The transfer shown requires about 105 days to complete, compared with
the required 5 days for a direct Hohmann transfer. However, the transfer requires only about 3.271 km/s
compared with a Hohmann transfer’s approximate 3.9 km/s.

The planar “Shoot the Moon” trajectory by Koon et al.3 was designed by decoupling the Hiten
Problem into two subproblems: the “Transfer Problem” and the “Capture Problem”. The Transfer Problem
studies the transfer from LEO to the vicinity of the Moon by first traveling into a region roughly 1 – 1.5
million km away from the Earth; the Capture Problem studies how to ballistically transfer into an orbit

7
temporarily captured by the Moon. Stable lunar orbits may also be accessed, but additional maneuvers are
required for their insertion.
The motion observed in the Transfer Problem strongly suggests that the invariant manifolds of two-
dimensional libration orbits, i.e., Lyapunov orbits, around EL1 or EL2 play a significant role in the ballistic
lunar transfers. Koon et al.3 showed that the invariant manifolds of Lyapunov orbits around EL1 or EL2
may indeed be used to construct a ballistic transfer from LEO to a lunar capture orbit using the following
method. Koon et al. recognized that the two-dimensional manifolds of a Lyapunov orbit topologically
separate the three-dimensional energy surface of the PCRTBP into an interior region and an exterior region.
This topological separation is significant because particles that travel in the interior region of the manifolds
behave very differently from particles that travel in the exterior region of the manifolds. A particle that is
temporarily captured by the Earth may escape if it is following a trajectory within the interior of the
unstable manifolds of a Lyapunov orbit about either EL1 or EL2. Similarly, all trajectories that become
temporarily captured by the Moon must pass through the interior of the stable manifold of a lunar
Lyapunov orbit about either LL1 or LL2.
The goal of the Transfer Problem is to depart from a LEO parking orbit and ballistically transfer into
a temporarily captured orbit about the Moon. Such a transfer may be constructed by exploiting the
dynamics of the invariant manifolds of Lyapunov orbits. A spacecraft may shadow the stable manifold of
the Lyapunov orbit to depart the Earth and then use the orbit’s unstable manifold to approach the Moon’s
orbit. One may be assured that such a trajectory will not escape the Earth’s vicinity by ensuring that the
spacecraft remains in the exterior region of the Lyapunov orbit’s manifolds. By properly phasing the
trajectory with the Moon, mission designers may arrange for the trajectory to intersect the stable manifold
of a desired LL2 Lyapunov orbit. If the trajectory is constructed to intersect the interior of the lunar orbit’s
manifold, then the spacecraft will freely transfer into a temporarily captured orbit around the Moon. If all
of these intersections occur in full phase space, then the transfer is entirely free.
A three-dimensional Shoot the Moon transfer is achieved in the next section using a slightly different
approach, but using the same dynamical systems theory.

3D SHOOT THE MOON TRANSFER

The two-dimensional Shoot the Moon transfer was very successful, although limited by its planar
requirement. A three-dimensional transfer would offer the same benefits as the two-dimensional transfer
along with additional advantages, including more options for the initial Earth parking orbit and the final
lunar orbit, better geometry for communication purposes, access to other regions of space, etc. This section
will introduce a three-dimensional version of the Shoot the Moon transfer.
The two-dimensional Shoot the Moon transfer was successful because the manifolds of periodic
orbits in the PCRTBP separated the energy surface into interior and exterior regions. The spatial CRTBP
has a six-dimensional phase space making its energy surface five-dimensional. Its periodic orbits, e.g. halo
orbits, have two-dimensional manifolds and its quasiperiodic orbits, e.g., Lissajous orbits, have three-
dimensional manifolds. Even three-dimensional surfaces cannot separate a five-dimensional energy
surface into useful regions. Consequently, the three-dimensional Shoot the Moon transfer cannot be
constructed in the same manner as its two-dimensional predecessor. Nevertheless, the invariant manifolds
of three-dimensional orbits in the CRTBP still indicate the flow in the system; the orbits may still be used
as staging orbits for a transfer. Theoretically, one may construct a three-dimensional Shoot the Moon
transfer by phasing the Moon such that the stable manifold of a lunar libration orbit intersects the unstable
manifold of a libration orbit about EL1 or EL2. If the stable manifold of the libration orbit about EL1 or EL2
intersects the Earth, then a Hiten-like trajectory may be constructed in the spatial system.

8
 The selection of the type of libration orbits for the staging orbits must be given careful consideration
due to the topology of the problem. An Earth staging orbit must be selected with the requirements that its
stable manifold intersects the Earth and its unstable manifold intersects the Moon’s orbit. The lunar staging
orbit must be selected with the requirement that its stable manifold intersects the unstable manifold of the
Earth’s staging orbit in full phase space. Only then can the transfer be free of deterministic maneuvers. In
a five-dimensional energy surface, it is much easier to intersect three-dimensional invariant manifolds than
two-dimensional manifolds. This suggests that quasiperiodic orbits such as Lissajous orbits have a better
chance of successfully intersecting their manifolds than periodic orbits such as halo orbits. In this paper, an
end-to-end solution was constructed using a Lissajous orbit around EL2 and a halo orbit around LL2.

The solution presented in this paper consists of the following. First, the spacecraft departs its initial
200-km LEO parking orbit on a trajectory that shadows the stable manifold of a Lissajous orbit about EL 2.
Because of the difficulty in viewing Lissajous orbits and their manifolds, a halo orbit with the same Jacobi
constant has been used for visualization purposes in the figures below. Second, the spacecraft transfers
from the stable manifold to the unstable manifold of the EL2 Lissajous orbit and approaches the Moon.
Finally, the trajectory is timed so that the stable manifold of a halo orbit about the Moon’s L 2 point
intersects the spacecraft as the spacecraft departs the EL 2 Lissajous orbit. Thus, the spacecraft ballistically
transfers from its LEO parking orbit onto a trajectory that either injects the spacecraft onto the LL2 orbit or
uses the manifolds of the LL2 orbit to guide it to a more desirable final orbit near or about the Moon. A
spacecraft on a halo orbit about the Moon’s L2 point may remain there as a science, telecommunication, or
support satellite; it may freely transfer to another unstable orbit near the Moon, such as a halo orbit about
LL1; it may freely transfer to a temporarily captured orbit about the Moon; or it may perform another
maneuver to transfer onto any type of conic orbit about the Moon, including equatorial to polar orbits at
any distance from the Moon.

Each step of this ballistic lunar transfer will now be discussed in more detail.

Earth Staging Orbits

The following requirements must be met to construct a proper Earth staging orbit for a three-
dimensional Shoot the Moon trajectory:

1. The orbit must be unstable;

2. The orbit’s stable manifold must intersect LEO, or another intermediate staging orbit;
3. The orbit’s unstable manifold must intersect the lunar staging orbit’s stable manifold.

There are several options that exist that meet these requirements, including halo orbits and Lissajous orbits.
Halo orbits are limited to a very small class of orbits, but they are easy to construct, manipulate, and
visualize; Lissajous orbits span a much broader region in the design space and thus provide better options
for real missions, but are harder to construct and visualize. The Shoot the Moon transfer constructed in this
paper implements a Lissajous staging orbit, but uses halo orbits to visualize the transfer connections.
Figure 6 shows four perspectives of the family of Northern halo orbits centered about the Sun-Earth L2
point. The family of Southern halo orbits is identical and has the same properties, except the orbits are
reflected about the x-y axis. Many of these orbits could be used in a lunar transfer depending on the
requirements of the mission. The specific orbit may only be chosen once one has identified the destination
of the spacecraft and the lunar staging orbit used. Then one can identify which Earth staging orbit’s
manifolds intersect the stable manifold of the lunar staging orbit.

9
 1.5

2 1

1.5
Moon’s 0.5
1
Orbit

z (x106 km)

y (x106 km)
0.5
0
0 EL2
-0.5
-0.5
-1

1 -1
0 151

-1 150
-1.5
y (x106 km) 149 149 149.5 150 150.5 151 151.5
x (x106 km)
x (x106 km)

2 2

1.5 1.5

1 1
z (x106 km)

z (x106 km)
0.5 0.5

0 0

Earth
-0.5 -0.5
EL2
-1 -1
149 149.5 150 150.5 151 151.5 -1.5 -1 -0.5 0 0.5 1 1.5
x (x106 km) y (x106 km)

Figure 6. Four perspectives on the family of Northern EL2 halo orbits.

Figure 7 shows four plots of an example halo orbit about the Sun-Earth L2 point and its stable
manifold. One can see that this stable manifold does intersect the Earth. Thus, a spacecraft could make a
single maneuver to transfer from a LEO parking orbit to a trajectory on this halo orbit’s stable manifold.
Similarly, Figure 8 shows four plots of the same halo orbit’s unstable manifold, showing that trajectories
exist that intersect the Moon’s orbit about the Earth. Thus, a spacecraft on, or near, the halo orbit may use
the orbit’s unstable manifold to guide it to intersect the Moon. The invariant manifolds of Lissajous orbits
with similar Jacobi constants also demonstrate the same properties, making them viable candidates for
Shoot the Moon staging orbits.

0.8

0.6

0.4

0.4
0.2
z (x106 km)

0.2
y (x106 km)

0
0
Earth EL2
-0.2 -0.2
-0.4
-0.4
0.5 151
-0.6
0 150.5
150 -0.8
-0.5
149.5
y (x106 km) x (x106 km) 149.2 149.4 149.6 149.8 150 150.2 150.4 150.6 150.8 151 151.2
x (x106 km)

10
 0.8
0.6
0.6

0.4
0.4

0.2 0.2

z (x106 km)
z (x106 km)
0 0
Earth EL2 Earth
-0.2
-0.2

-0.4
-0.4

-0.6
-0.6
149.4 149.6 149.8 150 150.2 150.4 150.6 150.8 151 151.2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
x (x106 km) y (x106 km)

Figure 7. Four perspectives of an example Northern EL2 halo orbit and its stable manifold. The stable manifold
intersects the Earth.

0.8

0.6

0.4
0.4
z (x106 km)

0.2 0.2

y (x106 km)
0
0
-0.2 EL2
-0.2
Earth
-0.4

-0.4
0.5 151
150.5 -0.6
0
150
-0.5 -0.8
149.5
y (x106 km) x (x106 km) 149.2 149.4 149.6 149.8 150 150.2 150.4 150.6 150.8 151 151.2
x (x106 km)

0.8
0.6
0.6

0.4
0.4

0.2 0.2
z (x106 km)
z (x106 km)

0 0

Earth EL2 Earth

-0.2
-0.2

-0.4
-0.4

-0.6
-0.6
149.4 149.6 149.8 150 150.2 150.4 150.6 150.8 151 151.2 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
x (x106 km) y (x106 km)

Figure 8. Four perspectives of an example Northern EL2 halo orbit and its unstable manifold. The unstable
manifold intersects the Moon’s orbit.

This section has demonstrated that a three-dimensional libration orbit about the Sun-Earth L 2 point
can be used as a staging orbit to transfer a spacecraft from a LEO parking orbit to a trajectory that
encounters the Moon. The next section explores lunar staging orbit options.

Lunar Halo Staging Orbits

This study has implemented halo orbits about LL2 as staging orbits for the three-dimensional Shoot
the Moon transfer, although numerous other orbits exist that would work as well. Halo orbits are easy to
construct and visualize; in reality quasi-periodic Lissajous orbits are more practical since they add an

11
additional degree of freedom to the design space. Figure 9 shows four perspectives on the family of
Northern LL2 halo orbits. Orbits in the Southern family are identical to Northern halo orbits except they are
reflected about the x-y axis. These orbits demonstrate the great variety of available options that one may
use as staging orbits about LL2 even in the family of halo orbits alone. All of these orbits are unstable, thus
a spacecraft may arrive onto one of these orbits for free by following that orbit’s stable manifold. The
spacecraft may then depart this staging orbit for a free transfer to a trajectory that intersects another desired
final lunar orbit.

60

80

40
60

40 20
z (x103 km)

y (x103 km)
20
0

0 Moon LL2
-20
-20

50 -40

0
440 460
-50 400 420 -60
360 380 360 380 400 420 440 460
y (x103 km)
x (x103 km) x (x103 km)

80 80

60 60

40 40
z (x103 km)

z (x103 km)

20 20

0 0

Moon Moon
-20 -20
LL2
360 380 400 420 440 460 -60 -40 -20 0 20 40 60
x (x103 km) y (x103 km)

Figure 9. Four perspectives on the family of Northern LL2 halo orbits.

Figure 10 shows four perspectives of an example LL2 halo orbit, along with its stable manifold. If a
spacecraft targeted a trajectory on this manifold, it would asymptotically approach the staging orbit. Thus,
if a spacecraft were able to transfer from the EL2 staging orbit’s unstable manifold, e.g. the manifold shown
in Figure 8, onto this LL2 halo orbit’s stable manifold, e.g., the manifold shown in Figure 10, then the
spacecraft will have achieved a ballistic transfer to this lunar orbit from LEO. By phasing the Moon
properly, one may align and intersect the two manifolds to accomplish this task.

If a mission’s goals were not to enter the staging orbit, but merely pass by it, then the spacecraft
could target a trajectory that shadowed this stable manifold instead of a trajectory that was on this
manifold. The stable manifold could then be used as an initial guess for a trajectory optimization tool to
converge on a desired trajectory.

12
 100

z (x103 km)
0

-100

1000
800
600
400 600
400
200
200
y (x103 km) 0 0
x (x103 km)

200 300

200
100
100
z (x103 km)

z (x103 km)
0 0

-100
-100

-200

-200
-300

-300 -400

-100 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 800 900 1000
x (x103 km) y (x103 km)

Figure 10. An example halo orbit about LL2 and its stable manifold in four perspectives.

If one propagates the stable manifold backward in time, i.e., in the process of intersecting it with the
EL2 orbit’s manifold, one notices that it quickly departs the Moon’s vicinity. Because the Sun plays such a
strong role in the motion of satellites near the Earth, trajectories on the lunar orbit’s stable manifold quickly
begin to be affected by the Sun. This study therefore switches from the Earth-Moon three-body system to
the Sun-Earth three-body system once the trajectories pass through the three-body sphere of influence. The
trajectory is then propagated in such a way that it continues to switch coordinate systems anytime it crosses
the 3BSOI boundary.

Lunar Capture Orbits from the LL2 Staging Orbit

The orbits shown in Figure 9, above, indicate available options for staging orbits at the Moon. These
orbits may provide good locations for communication satellites, relay satellites, or space stations. If a
spacecraft needs to transfer to a lower lunar orbit, then the spacecraft may follow the halo orbit’s unstable
manifold (for free) toward the Moon, intersect its final lunar orbit, and perform a capture maneuver. To
demonstrate this, Figure 11 shows an example lunar halo staging orbit about LL 2 and its unstable manifold.
A spacecraft on this halo orbit could depart along any one of these trajectories. These trajectories fly by the
Moon at different radii and inclinations; thus, many different final lunar orbits are accessible from this
staging orbit. Figure 12 shows one such trajectory that a spacecraft could take to depart the lunar halo
staging orbit. If a maneuver were performed at perilune the spacecraft would be captured into a stable orbit
about the Moon, thereby completing the end-to-end trajectory solving the Spatial Hiten Problem.

13
 LL1 LL2

Moon

30

20 20

10 10

0 0
LL1 Moon LL2
-10 -10
Moon
z (x103 km)

z (x103 km)
-20 -20

-30 -30

-40 -40

-50 -50

-60 -60

340 360 380 400 420 440 -50 -40 -30 -20 -10 0 10 20 30 40 50
x (x103 km) y (x103 km)

Figure 11. An example halo orbit about LL2 and its unstable manifold in four perspectives.

50
20
40

30
0

20
Moon
z (x103 km)

-20
10
y (x103 km)

-40 0
LL1 Moon LL2
-10
-60
-20
50
-30

0 -40

440 -50
-50 400 420
360 380 320 340 360 380 400 420 440
y (x10 3 km) 340
x (x10 3 km) x (x10 3 km)

20 20

10 10
Moon
0 0
Moon
-10 -10
z (x103 km)

z (x103 km)

-20 -20

-30 -30

-40 -40

-50 -50

-60 -60

360 380 400 420 440 460 -50 -40 -30 -20 -10 0 10 20 30 40 50
x (x10 3 km) y (x10 3 km)

Figure 12. A trajectory from Figure 11, demonstrating an example destination for a spacecraft on a 3D Shoot
the Moon trajectory. The trajectory is shown in four perspectives.

14
Additionally, since each halo orbit shown in Figure 9 has a different unstable manifold, the three-
dimensional Shoot the Moon transfer can access nearly any lunar orbit. Figure 13, below, shows some
available options for the radius and inclination of lunar orbits that may be accessed by Northern LL 2 halo
orbits. The highlighted points are those that are accessible from the example halo staging orbit shown in
Figure 11. Southern halo orbits can access the same set of lunar orbits except with a negative inclination.

Figure 13. A plot showing the radius and inclination of lunar orbits that are accessible from the Northern LL2
halo staging orbits. The highlighted points are accessible from the example halo orbit shown in Figure 11.

Full LEO to Lunar Halo Transfer

A full three-dimensional Shoot the Moon transfer involves two staging orbits: one in the Sun-Earth
system and one in the Earth-Moon system. The orbits selected are members of full families of orbits,
where the family includes orbits that share similar features and whose orbital parameters vary continuously
from one end of the family to the other16. Thus, if a Shoot the Moon transfer is identified, then it will also
be one member of a full family of such transfers. This study has constructed such a family of transfers.
For brevity, only one example of such a transfer will be presented here. The full family of transfers will be
discussed in a future paper.

Figure 14 shows the first portion of the three-dimensional Shoot the Moon transfer in four
perspectives. The spacecraft is launched from a 200-km low-Earth orbit, travels outward toward EL 2 along
a trajectory that shadows the stable manifold of an orbit about EL2, skims the periodic orbit and then travels
toward the Moon. The spacecraft’s motion shadows the manifolds of a Lissajous orbit, however, the
Lissajous orbit’s manifold is very difficult to visualize. Therefore, the plots shown in Figure 14 include the
stable manifold of the halo orbit with the same Jacobi constant to represent the nearby Lissajous manifolds.
The spacecraft’s motion skims this manifold as well, although not quite as closely as the Lissajous orbit’s
manifold.

15
 0.8

0.6

0.4
0.5
0.2

y (x106 km)
z (x106 km)
0
0
Earth
-0.2

-0.5 -0.4

-0.6
0.5 151

0 150.5 -0.8
150
-0.5
149.5 -1
149 149.5 150 150.5 151
y (x106 km) -1 149 x (x106 km)
x (x106 km)

0.4

0.4
0.2
0.2

z (x106 km)
z (x106 km)

0 0

-0.2
-0.2

-0.4
-0.4
-0.6
149 149.5 150 150.5 151
x (x106 km) -0.6
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8
y (x106 km)

Figure 14. The 3D Shoot the Moon trajectory plotted alongside an EL2 halo orbit’s stable manifold.

Figure 15 shows four perspectives of the same transfer trajectory, but this time with the EL 2 halo
orbit’s unstable manifold plotted. One can see that the trajectory shadows the unstable manifold as it
approaches the Moon.

0.8

0.6

0.5 0.4

0.2
z (x106 km)

y (x106 km)

0
0

-0.2
-0.5

-0.4 Earth
1

0.5 151 -0.6

0 150.5
-0.8
150
-0.5
149.5
-1
y (x106 km) -1 149 x (x106 km) 149 149.5 150 150.5 151
x (x106 km)

16
 0.4
0.4

0.2
0.2

z (x106 km)

z (x106 km)
0 0

-0.2 -0.2

-0.4
-0.4
-0.6
149 149.5 150 150.5 151 -0.6
x (x106 km) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
y (x106 km)

Figure 15. The 3D Shoot the Moon transfer plotted alongside an EL2 halo orbit’s unstable manifold.

Figure 16 shows the same four perspectives of the three-dimensional Shoot the Moon transfer plotted
alongside the LL2 halo orbit’s stable manifold. One can see that the transfer lies on the manifold, indicating
that the spacecraft has injected into the LL2 halo orbit and has become temporarily captured by the Moon.
If the mission objective were to achieve a final inclined low-lunar orbit and not a libration orbit, then the
trajectory would not lie directly on the manifold, but would shadow the manifold. It would skim the LL 2
orbit’s stable manifold and transfer to the LL2 orbit’s unstable manifold before intersecting the final lunar
orbit.

Figure 17, below, shows the top-down perspective of the entire three-dimensional Shoot the Moon
transfer with all three manifolds displayed.

0.5

0.4

0.3

0.2 0.2
z (x106 km)

y (x106 km)

0.1
0
0
-0.2
-0.1 Earth
0.5
150.2 -0.2
150
0 149.8 -0.3
149.6
149.4
149.2
-0.4
Moon
y (x106 km) -0.5 x (x106 km) -0.5
149.2 149.4 149.6 149.8 150 150.2
x (x106 km)

0.2
0.2

0.1 0.1
z (x106 km)

z (x106 km)

0 0

-0.1 -0.1

-0.2
149.2 149.4 149.6 149.8 150 150.2 -0.2
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
x (x106 km)
y (x106 km)

Figure 16. The 3D Shoot the Moon transfer plotted alongside the LL2 halo orbit’s stable manifold.

17
 1

0.8 WUEL2

0.6

0.4

0.2
y (x106 km)

0
EL2
-0.2
Earth

-0.4 Moon

-0.6 WSLL2

-0.8

-1
149 149.5 150 150.5 151
6
x (x10 km)
WSEL2

Figure 17. The 3D Shoot the Moon transfer plotted alongside the stable and unstable manifolds of the EL2 halo
staging orbit and the stable manifold of the LL2 halo staging orbit.

The total cost of the trajectory shown in the previous figures is only about 3.261 km/s, executed in a
single maneuver at the LEO departure point. The inclination of the LEO parking orbit in this trajectory is
approximately 25°. After that single maneuver, the only other maneuvers required are station-keeping
maneuvers, which may be arbitrarily small based on other mission constraints. The cost is heuristically
cheaper than Hohmann-type transfers to inclined lunar orbits, but the quantitative comparison has not been
completed here since so many factors play in the construction of an inclined Hohmann transfer. It is
interesting that most features in the three-dimensional transfer are very similar to the two-dimensional
transfer computed by Koon et al. The duration and maneuver magnitude of the three-dimensional transfer
are approximately the same as the duration and maneuver magnitude of a similar two-dimensional transfer.
This is summarized in Table 1.

Table 1. A comparison of three lunar transfer types from a 200-km LEO

parking orbit to an orbit temporarily captured by the Moon.
Transfer Type Hohmann* 2D Shoot the Moon 3D Shoot the Moon
ΔV 3.73 – 3.95 km/s 3.27 km/s 3.26 km/s
Inclination range of
0° 0° Any†
LEO parking orbit
Inclination range of final
0° 0° Any†
lunar orbit
Transfer duration 3.9 – 5.2 days 105 days 105 days
*
The case of the Hohmann transfer includes final lunar orbits from 100-km out to L 1. †The inclination range includes
the full family of orbit transfers, although this paper presented one that departed from a 25° LEO orbit.

18
Validation
The decoupled four-body system only approximates the motion of the full four-body system. More
realistic models may be used to validate the results constructed in the decoupled four-body system. This
section reproduces the Shoot the Moon transfer trajectory in the bicircular model.

Bicircular Model
The bicircular model of the Sun-Earth-Moon system is another model of the four-body system. In
the bicircular model, the Earth and Moon orbit about their barycenter in circular orbits in exactly the same
manner as in the circular restricted three-body system. However, the Earth-Moon barycenter and the Sun
also orbit each other in circular orbits. Thus, the bicircular model may be described as the Earth-Moon
circular restricted three-body system with the addition of the Sun orbiting the barycenter in circular motion.
This paper restricts all three primary bodies to the same plane; however, the bicircular model may be
adjusted to account for the inclination of the Moon’s orbit about the Earth with respect to the ecliptic. For
more information about the bicircular model, see Gómez et al.17

The Shoot the Moon transfer constructed in the bicircular model appears nearly identical to the
transfer shown above, and it is heuristically identical, but the two transfers are quantitatively slightly
different. The transfer modeled in the bicircular model only requires a ΔV from the 200-km LEO parking
orbit of approximately 3.175 km/s instead of the 3.261 km/s required by the decoupled four-body model.
In the bicircular model the trajectory departs the Earth from a LEO parking orbit with an inclination of
approximately 10.3°. A top-down perspective of this transfer is shown in Figure 18.
o o
φ = 0.201700, θ0 = 44.209000 , hp = -5638.547 km, ∆V = 9.6043 km/s, inc = 31.79

0.5 Moon’s
Earth’s Orbit
Orbit
y (x106 km)

0
EL1 EL2

-0.5

-1

148 148.5 149 149.5 150 150.5 151 151.5

x (x106 km)

Figure 18. The 3D Shoot the Moon transfer constructed in the bicircular model.

19
CONCLUSIONS AND FUTURE WORK

This paper has demonstrated how invariant manifold theory may be used to construct and understand
three-dimensional ballistic lunar transfers. A specific transfer has been constructed that transfers from a
200-km LEO parking orbit into a halo orbit about LL 2 using only a single 3.26 km/s maneuver at the LEO
injection point. A spacecraft on such a transfer could remain on the halo orbit, freely transfer to another
libration orbit, freely transfer to a temporarily captured orbit about the Moon, or perform another maneuver
to inject into any lunar orbit. The transfer implements three-dimensional libration orbits as staging orbits in
the Sun-Earth and Earth-Moon three-body systems. Other orbits could be used to produce similar results.
The advantages of the three-dimensional approach, compared with the two-dimensional work produced by
Koon et al., include access to inclined lunar orbits, the option to use inclined LEO parking orbits, access to
other regions of space in the Earth’s neighborhood, and better communication geometry. The Shoot the
Moon transfer requires less energy than conventional Hohmann transfers, but requires a much longer
transfer time. Such low-energy transfers are useful for cargo transport and robotic missions to lunar orbits
or to the surface of the Moon. The lower ΔV cost permits larger payloads and/or smaller rockets to reach
the Moon.

The approach used in this paper is robust enough to identify families of ballistic lunar transfers,
including transfers that use other types of unstable orbits as staging orbits. These families will be explored
and presented in future papers. Further analysis of the structure of the invariant manifolds of three-
dimensional orbits in the spatial CRTBP may provide additional understanding of these lunar transfers,
which may present alternative approaches for the construction and analysis of ballistic lunar transfers. The
methods presented by Gómez et al.18, for example, may offer such an alternative. These alternative
approaches will be explored in future papers. The goal of this research is to provide mission designers the
tools needed to quickly construct and evaluate the performance of these lunar transfers for their own unique
missions.

REFERENCES
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IOM 312/90.4-1731-EAB, June 15, 1990.
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 10. Hénon, M., “Exploration Numérique du Problème des Trois Corps, (II), Masses Egales, Orbites
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ACKNOWLEDGEMENTS

This work has been completed under funding by a National Defense Science and Engineering
Graduate (NDSEG) Fellowship sponsored by the Deputy Under Secretary of Defense for Science and
Technology and the Office of Naval Research.

21