Analyzing Prograde Orbits in the EarthMoon ThreeBody System

1
Analyzing Prograde Orbits in the Earth-Moon
Three-Body System
Jeffrey S. ParkerColorado Center for
Astrodynamics ResearchUniversity of Colorado at
Boulderparkerjs_at_colorado.eduhttp//www-ccar.colo
rado.edu/parkerjs/Home.htm
Martin W. LoJet Propulsion LaboratoryCalifornia
Institute of TechnologyMartin.Lo_at_jpl.nasa.govhtt
p//www.gg.caltech.edu/mwl/
Abstract The nature of unstable
three-body orbits has only recently been studied
with any practical application in mind. Yet,
even with incomplete knowledge of the orbital
possibilities in the system, many highly
successful mission designs have been constructed
that implemented unstable orbits. The focus of
this study is to explore the practical potential
of another family of unstable orbits in the
Earth-Moon three-body system, namely periodic
distant prograde orbits.
Mission Design Opportunities The
manifolds of unstable orbits help to reveal
mission design options in the three-body system.
The manifolds shown in Figure 2, among other DPO
manifolds indicate that if one can arrive onto a
particular DPO, one can use the dynamics in the
system to then transfer for free to any of the
five Lagrange points in both the Earth-Moon and
Sun-Earth systems, or to a trajectory that
intersects the earth or the moon. Koon et al.
(2000) studied similar transfers using libration
orbits about the Sun-Earth L1 and L2 points.
Figure 3
shows such a trajectory that departs
from a low-Earth orbit, passes near a Sun-Earth
halo orbit, and arrives at the moon without
requiring a second burn. Similar designs could
be produced implementing the unstable periodic
prograde orbits shown here. Figure 4 shows the
stable and unstable manifolds for a specific
Sun-Earth DPO that could easily be used to
transfer from LEO to the moon with only the
single LEO burn.
Distant Prograde Orbits The distant
prograde orbits (DPOs) explored in this study
orbit the moon in a prograde fashion with
significant gravitational perturbations by the
earth. Several types of these orbits are shown
in the figures to the right, including central
orbits (green) and unstable orbits (red). These
orbits are all related and can be classified
using a bifurcation plot in phase space, as shown
in the DPO Design Space plot to the right.
Arrows indicate where these orbit types can be
found in phase space. Currently the design space
is limited to orbits whose velocity is purely
orthogonal to the x-axis as they cross the
x-axis. Even with this limitation, many
different classes of DPOs have been identified
using the phase space representation.
The red unstable DPOs shown in the Design
Space are particularly interesting for mission
designers because their stable and unstable
manifolds provide invariant tubes for particles
and spacecraft to travel along. These tubes are
abstract constructs that are very useful as
tools to
understand
methods for which interplanetary material is
transported in the solar system, as well as
within the host three-body system. DPO tubes are
therefore contributors to the Interplanetary
Superhighway much like the well-known libration
orbit tubes (see the artist conception shown
above).
Invariant Manifolds The set of
trajectories that a particle or spacecraft could
use to arrive onto an unstable orbit defines that
orbits stable manifold. Because the orbit is
unstable, a small perturbation could cause the
particle or spacecraft to depart from the orbit
and follow a trajectory in that orbits unstable
manifold. The manifold, as well as the orbit
itself, are constructs that only contain
structure in the rotating frame. Figure 1,
right, demonstrates this by comparing a periodic
DPO in an Earth-centered inertial frame versus a
Moon-centered rotating frame. The structure is
much more obvious in the rotating frame.
Orbital transfers are found when one orbits
stable manifold intersects another orbits
unstable manifold. Therefore it is useful to
explore the characteristics of both manifolds for
each orbit. The stable and unstable invariant
manifolds for a typical class of unstable DPOs in
the Earth-Moon system are shown in Figure 2. One
can see that the stable manifold (blue)
intersects many regions in the system regions
near the moon, near the Earth-Moon Lagrange
points, near the Sun-Earth Lagrange points, and
near the earth. Hence, there are many available
options that a particle or spacecraft could take
to arrive onto this periodic orbit. Furthermore,
the unstable manifold (red) intersects the same
regions as the stable manifold. Thus, once a
particle is on or near this DPO, it can reach any
of those regions in the system. The implication
for free transfers in the Earth-Moon system is
great. The third plot shows a closer view of the
unstable manifolds trajectories near the moon,
suggesting numerous possibilities to land or go
into orbit about the moon.
Conclusions Unstable distant prograde
orbits produce manifold tubes that can be used in
practical mission designs much like those
previously included in the Interplanetary
Superhighway. The manifolds of the DPOs in this
study imply a great potential for free transfers
between orbits in the vicinity of the earth, the
moon, the Sun-Earth Lagrange points and the
Earth-Moon Lagrange points.
References Howell, K.C., Three-Dimensional,
Periodic, Halo Orbits, Celes. Mech., 32, 53,
1984. Koon, W.S., Lo, M.W., Marsden, J.E.,
Ross, S.D., Shoot the Moon, Spaceflight
Mechanics 2000, AAS, 105, Part II, 1017-1030,
2000. Ocampo, C.A. and Rosborough, G.W.,
Transfer Trajectories for Distant Retrograde
Orbiters of the Earth, AAS, 180,
1993. Szebehely, V., Theory of Orbits, The
Restricted Problem of Three Bodies, Academic
Press, 1967.
Acknowledgements This work has been
completed under partial 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 and by the
National Aeronautics and Space Administration
through Jet Propulsion Laboratory, California
Institute of Technology. We wish to
express sincere gratitude for Dr. George Born,
the director of the Colorado Center for
Astrodynamics Research, who has supported us
during this research.