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Mapping the HIGHways of Space
Posted by:
Heather Cohea
Graduate Student
in Aerospace Engineering
University of Colorado—Boulder
12 December, 2001
above photo from HST website
Abstract
Libration points are areas of space where the gravitational pull from two or more bodies, plus the
centrifugal force from their rotation balance. Therefore, there are only very subtle gravitational
tugs from distant planets or asteroids that tend to disturb a craft orbiting a libration point. There
are also complicated gravitational fields in space that can form two-dimensional "surfaces"
around libration points. These surfaces extend out into space and hold slight curves and bends
that can guide a spacecraft to a different libration point. It is also plausible that a spacecraft may
transfer from surface to surface to get to another target point. This paper will discuss the uses of
these gravitational sheets and propose a mission that could find their locations.
Background
What are Lagrange Points?
In the vicinity of two orbiting masses there are positions where a third, smaller mass can orbit at
a fixed distance from the larger masses. The Lagrange Points mark positions where the
gravitational pull of the two large masses precisely equals the centripetal force required to rotate
with them. Also known as "libration points" or "L-points," there are 5 positions where a light
third body can sit "motionless" with respect to the two heavier bodies.
Figure 1 below depicts the area where Lagrange Points can be found in the vicinity of the two
orbiting bodies.
Figure 1: Image from the MAP website
Of the five L-points, two are stable and three are unstable. The L4 and L5 points are home to
stable orbits so long as the mass ratio between the two large masses exceeds 24.96. This
condition is satisfied for both the Earth-Sun and Earth-Moon systems, and for many other pairs
of bodies in the solar system. L4 and L5 form the apex of two equilateral triangles that have the
large orbiting masses as their vertices. The unstable points, named L1, L2, and L3, lie on the line
through the two heavy masses (e.g. sun and earth). L1 is between the sun and earth, L2 is antisunward of the earth, and L3 is on the far side of the sun.
How are they located?
Joseph-Louis Lagrange, an Italian-French mathematician, discovered the five
special points. This web link has a PDF version of Lagrange's derivation
(168K PDF file, 8 pages).
One way to envision the
Lagrange derivation is
to observe the objects
from a frame of
reference that rotates
with the system. In this
frame, the forces exerted
on a body at rest can be
derived from an
effective gravitational
potential. When the
contours are closer, the
forces are stronger, and
when the contours are
farther the forces are
weaker.
In the contour plot in
Figure 2, highs are
yellow and lows are
purple. L4 and L5
Figure 2 from the MAP website
correspond to maxima
and L1, L2, and L3
correspond to saddles (i.e. points where the potential is curving up in one direction and down in
the other). This suggests that satellites placed at the Lagrange points will have a tendency to go
astray. This is true for L1, L2 and L3, but not for L4 and L5. When a satellite near L4 or L5
starts to "roll-off the hill" it picks up speed, but the Coriolis force sends the satellite into a stable
orbit around the Lagrange point (Cornish).
Restricted Three Body Problem
The following information was obtained from an article by Amara Graps for the Solar Center at Stanford.
There currently exists no complete analytical solution to the problem of three orbiting bodies
which are under the influence of only their mutual gravitational attraction. The restricted threebody problem makes the assumptions that two bodies of finite mass revolve around one another
and a third body of infinitesimal mass moves in their field. The third object is assumed to have
no measurable affect on the motion of the two other bodies. The orbits of the two finite mass
bodies conform to the solution to the two-body problem, and therefore, the puzzle is reduced to
analyzing the motion of the infinitesimal particle in the field of the two co-orbiting bodies.
Further, assuming the two bodies of large mass are in circular orbits about one another simplifies
the problem to the "circular restricted three-body problem."
Setting the center of mass of the two finite objects as the origin of the circular restricted threebody problem and assuming the axes rotate with constant angular velocity, an energy integral
falls out of the constants of integration. It is named "Jacobi's integral" after the German
mathematician Carl Gustav Jacobi, and it represents the conservation of energy in the rotating
system. (NOTE: The complete solution to the restricted three-body problem requires five more
integrals, but finding these has not been accomplished.)
There are five fixed points of equilibrium in the Circular Restricted Three-Body Problem that
can be found using the Jacobi Integral, where all components of position and velocity vectors for
the infinitesimal mass equal zero. These are the Lagrange Points.
The following website shows another derivation of the distances from Earth to the L1, L4 and L5
points.

L1:
derivation, see numbers 34a), 34b) and 34c).

So, What’s the Big Deal About L-Points?
Naturally Occurring Inhabitants of L-points
Because they are stable areas, mass naturally tends to accumulate about L4 and L5. Objects
found orbiting at the L4 and L5 points are often called Trojans after the three large asteroids
Agamemnon, Achilles and Hector that orbit in the L4 and L5 points of the Jupiter-Sun system.
(According to Homer, Hector was the Trojan champion slain by Achilles during King
Agamemnon's siege of Troy). There are hundreds of "Trojan" asteroids at stable Lagrange points
in the solar system. But although most orbit with Jupiter, others orbit with Mars, and several of
Saturn's moons have Trojan companions.

The asteroid 5261 Eureka is a "Mars Trojan", occupying one of the stable Lagrange
points of Mars' orbit around the sun.

At the Lagrange point 60 ahead of Saturn's moon, Dione, there is a tiny moon called
Helene..

No large asteroids have been found at the Trojan points of the Earth-Moon or Earth-Sun
systems. However, in 1956 the Polish astronomer Kordylewski discovered large
concentrations of dust at the Trojan points of the Earth-Moon system. They are about 4
times as big as the Moon.
Science
If it were possible to suspend a spacecraft motionless in space relative to the body it orbits, If a
satellite could remain stationary with respect to the body it orbits, much could be learned about
the dynamic space environment from observations of changes at that one fixed spot. Examples
include observing changes in magnetic fields and particle flows. Until recently, many thought
that it could not be done because from our understanding of celestial mechanics it is intuitive to
assume that in order to stay up and resist gravity, a satellite must be in constant motion and must
follow a prescribed orbit. But the Lagrange points provide a natural resistance where gravity and
centripetal force balance and this allows satellites to remain "motionless" with respect to the
larger orbiting bodies. This provides unique opportunities for research. For example, the L1 point
of the Earth-Sun system affords an uninterrupted view of the sun from a vantage point free from
the influence of Earth's magnetic field. The L2 point could be used as an observational platform
for satellites that peer into the outer expanses of the solar system.
Also, only small amounts of fuel are required to maintain a spacecraft about the L-points. Thus,
spacecraft placed in such regions can gather data over a longer period of time than other
missions, such as flybys, and can collect data from various regions of space, without being
limited to the space near planets.
Effectively Utilizing Lagrange Points
Have There Been Any Missions Around Lagrange Points?
The L1 point is a very good position for continual monitoring of the sun and advanced
monitoring of the solar wind because the solar wind reaches L1 about one hour before reaching
Earth. The International Sun-Earth Explorer-3 (ISEE-3) was launched in 1978 and it conducted
such observations for several years before moving to different locations for more science
missions. In November 1994, WIND was launched towards that position and was originally
scheduled to be stationed in an orbit about the L1 point, but later was sent on a more extended
mission in a "flower petal" orbit about Earth. L1 is now the home of SOHO, the Solar and
Heliospheric Observatory and ACE (Advanced Composition Explorer).
Figure 3 from the MAP website
NASA put the MAP (Microwave Anisotropy
Probe) satellite into the Lagrange point L2
using the orbit depicted in Figure 3. MAP
will study bumps in the cosmic microwave
background (CMB). The L2 Lagrange point
offers an ideal location from which to carry
out CMB observations because of its
distance of 1.5 million km from Earth. Thus,
it affords great protection from the Earth's
microwave emission, magnetic fields, and
other disturbances. It also provides for a very
stable thermal environment and near 100%
observing efficiency since the Sun, Earth,
and Moon are always behind the instrument's
field of view (MAP website).
Figure 4 from the MAP website
Also, NASA has chosen the L2 point as the
future site of a large infrared observatory, the Next Generation Space Telescope.
Problems
Rocket Propulsion System Required
Theoretically, a spacecraft placed at a libration point would stay there indefinitely. However, the
very subtle gravitational pull from distant planets or an asteroid could disturb a craft enough to
cause it to drift away. Thus, a spacecraft that is to be stationed at L1, L2 or L3 must have an
engine. These positions are unstable on a time scale of approximately 23 days, and when the
spacecraft slips away, it will continue to drift further. Thus, regular orbital corrections are
required. Furthermore, for the example of the L1 position, currently, the most economic way of
traveling to L1 is through a gravity-assist maneuver with the Moon. Those maneuvers require
on-board propulsion, and much fuel, as does the final approach to L1.
Another problem is that these orbits are extremely complicated and it is very difficult plot
trajectories to them. According to Martin Lo, a mission designer at NASA's Jet Propulsion
Laboratory in Pasadena, California, "they look like a line drawn around the edge of a Pringle's
potato chip," (Taubes). The "halo" orbits around Lagrange points are much more complex than
the orbit of a planet around the sun. This is why they are often referred to as orbits 'near' a
libration point instead of 'around' a libration point. Space missions are currently planned by
calculating a specific trajectory for each new mission. This is a complicated mathematical
process that must be repeated for every new mission.
Proposed Solutions
Professor Kathleen Howell at Purdue University in West Lafayette, Indiana has found that the
complicated gravitational fields in space can form two-dimensional "surfaces" around libration
points which extend out into space.
Figure 4 from Purdue Univ. Faculty website
"These surfaces, some of which pass relatively close to the Earth, contain gentle curves and
bends, like the surface of a sheet billowing in the wind. Ms. Howell believes that if you can get a
spacecraft anywhere on the sheet, it will naturally get into orbit around a particular libration
point, without having to use fuel to steer the craft," (Siegfried). Thus, a spacecraft could "surf"
along gravitational sheets to a Lagrange Point "like a marble rolling in a funnel," (Siegfried).
Furthermore, it is plausible that a spacecraft could "shift from one sheet to another to get to a
destination, not necessarily a libration point," (Siegfried).
There are many benefits to using these sheets for multiple mission objectives.

Once you find a sheet, you don't have to perform the calculations to find it again. Thus,
you cut down on complicated orbital design calculations.

They would be good for "low thrust" missions.

Although travel time would be longer, low thrust missions need less powerful engines
that require less fuel and so smaller spacecraft could be used.

Lowering the cost of planetary and exploratory missions could be realized because one
launch vehicle could transport several spacecraft to a particular libration point where they
would then disperse to different locations.

Jupiter or Saturn missions, for instance, could use local the surfaces to explore those
planets' neighborhoods and spacecraft could ride them from one moon to another in
weeks, with virtually no fuel.
In fact, Howell proposes to provide mission designers with a complete database of surfaces,
enabling them to click on the starting and ending points of a mission and have the computer
quickly and automatically plot a trajectory using the sheets. Howell and her graduate students are
looking for such surfaces in the solar system, determining where they go and then plotting them
on a computer.
Methodology
A large amount of the information in this section was obtained from an article by: Gary Taubes in the June, 1999
issue of Discover magazine.
Rather than try to calculate individual trajectories as though she were planning a space mission,
Howell used computers to calculate tens of thousands of trajectories whose initial conditions
differed only slightly. With one set of initial conditions, she would get a single trajectory. With,
say, ten sets of slightly different initial conditions, she might get not just ten lines in space but
ten lines that all lay on the same curvaceous surface, known in the lingo of mathematics as a
manifold. Because there is an infinite number of starting points for a spacecraft and an infinite
number of starting velocities, there is also an infinite number of manifolds.
Any group of three or more bodies will interact to create manifolds with bizarre, albeit subtle,
gravitational effects. For instance, a spacecraft nudged off a libration point in one direction might
float into space, following the curve of the manifold, while if it went off in another direction, it
would float right back to the libration point. Similarly, there were manifolds that would drop a
spacecraft onto a halo orbit, and other manifolds that would lift it off and drop it onto a different
halo orbit. If you could find the right manifolds, Howell believed, you could place your
spaceship anywhere you liked in the neighborhood of the Earth, sun, and moon. This sort of
design process was used to plan the trajectory for the Next Generation Space Telescope,
scheduled to replace the Hubble Space Telescope in eight years and is being investigated for use
in other near earth missions. Also, Mr. Lo and his Caltech collaborators have taken to calculating
libration points, halo orbits, and their attendant manifolds for all the planets in the solar system.
This technique, while an important first step, is virtually endless in computation because the
possibilities for solutions are infinite based on the initial conditions of the space vehicle. Also,
the solutions are calculated based on uses for a particular mission. Because each new mission has
unique objectives and requirements for trajectories, the manifolds found may not be the
dominant gravitational funnels in the area. Also, the technique is piecemeal meaning that there
will be many holes in the mapping of the manifolds that a particular new and unique mission
requires. There is a solution that could be used in conjunction with the computational technique
that might allow for a more useful database.
Proposed Advancement
Mapping Space HIGHways
Figure 6 from HST website
Figure 5 from Heavens Above
Instead of continually running computer code over
millions of calculations in order to get a new trajectory for each unique mission until all
manifolds are mapped, a spacecraft could be launched that is dedicated to mapping the space
HIGHways, or the manifold surfaces between Libration Points. Using a suggestion stated
previously in this report, one launch vehicle could travel to a Lagrange Point over existing
trajectories and drop off multiple small spacecraft to disperse. The spacecraft would each be
equipped with a transmitter and follow the dominant manifolds or surfaces without further need
of fuel. By tracking the trajectories of each tiny transmitter along the surfaces, the dominant
manifolds of the region would be mapped.
Figure 7 from Heavens Above
It is reasonable to believe that the
tiny spacecraft would follow the
HIGHways that have the most
probability of being pursued by other
objects. Thus, the most important
HIGHways would be mapped first and
entered into Howell’s database for
future mission planning. Later, more
spacecraft could be launched with the
capability of making small trajectory
adjustments in order to shift from one
HIGHway to another. In this way,
intersections of manifolds could be
plotted and entered into Howell’s
database.
There are several problems associated with this scenario that should be worked if it is to become
viable. One is the problem of diminishing signal return as the spacecraft travels further and
further from Earth. Research is currently being conducted to find ways to have efficient deep
space communication. Another solution is to use GPS satellites as relays and signal boosters, or
to set up another network of relay satellites.
Also, the number of manifolds available based on an infinite number of initial velocities even
given a starting location is still an issue. A couple of questions to ask are “is there an upper limit
to the initial velocity that would simply cause the spacecraft to shoot away from the L-point and
any manifold surface?” and “is there a lower limit to the initial velocity that would simply not be
enough to break free from stable Lagrange points?” The answers to these questions may offer
information to limit the number of possible manifolds.
Finally, another issue that must be resolved is the question of what to do with the exploratory
space vehicles once they have concluded with the mission. Leaving them as space debris would
be irresponsible but requiring some sort of de-orbit would require larger spacecraft and more
fuel. It is plausible that a small amount of fuel could be reserved to put them on a manifold
headed for a stable L4 or L5 point where they would be trapped for retrieval at some unspecified
later date.
Conclusions
Aside from the fact that they are fascinating on an intellectual level, Lagrange points and space
HIGHways are alluring in that they provide numerous possibilities for inexpensive deep space
missions. The problem of finding them seems immeasurably large until a method of reducing
the choices emerges. The method of tracking tiny space vehicles is concept that could make
mapping of space manifolds easier and quicker if the problems stated above are tackled and it is
used in conjunction with the computational method currently being employed by Howell and Lo.
References:

Baez, John, http://www.math.ucr.edu/home/baez/lagrange.html "Lagrange Points," 16 June 2000.

Bennett, Charles, responsible NASA Official, http://map.gsfc.nasa.gov/m_mm/ob_techorbit1.html "The
Lagrange Points."

Graps, Amara, http://solar-center.stanford.edu/FAQ/QL1.html "How is SOHO Orbit Location (The L1
Point) Calculated?" 22 July 1997.

O’Neill, Gerald K. "The Colonization of Space," Physics Today, Sept 1974 p32

Siegfried, Amanda, http://www.purdue.edu/UNS/html4ever/9702.Howell.space.html , "New ‘Wave’ In
Space Exploration May Be Coming Soon," Feb 1997.

Stern, David P., and Peredo Mauricio, http://www-spof.gsfc.nasa.gov/Education/wlagran.html, "#21
Lagrangian Points," 9 Jan 2001.


Taubes, Gary, http://www.findarticles.com/cf_dls/m1511/6_20/55926787/p1/article.jhtml ,Discover
Online, "Surfing the Solar System," June 1999.
http://members.aol.com/cclinker/l5data2.htm "The Three Body Problem-Lagrangian Equilibrium Points."