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If you were an estate agent selling hot properties in the Solar System, what
features would catch your eye? The most obvious ones are planets, moons,
asteroids and perhaps comets鈥攖he elements of the cosmic landscape that
correspond to material objects. But the prime real estate in the solar system is
far more nebulous: empty space. What makes agents drool when it comes to the
cosmic property market鈥攎uch as in the terrestrial one鈥攊s location.
Not location in the conventional landscape of matter, but in the ever-shifting
landscape of gravitation and time.
The desirability of certain routes or orbits has been known for some time.
Right now hundreds of telecommunications satellites encircle the Earth in one
specific orbit, roughly 36 000 kilometres above the equator. In this
orbit鈥攁nd only this one鈥攖he speed of the satellite matches the
rotation of the Earth precisely, so the satellite remains fixed relative to the
ground.
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What is new is the rapidly increasing catalogue of ever more exotic orbits,
fuelled largely by developments in the mathematical techniques needed to
distinguish them from the apparently identical empty space nearby. The mere
existence of many a cosmic des res was unsuspected even a few years ago, and
without some state-of-the-art mathematical concepts and techniques their
desirability would never have been recognised.
Raising the dead
The first time a truly exotic orbit showed up was around a decade ago, when a
bunch of NASA engineers got a bright idea about recycling satellite ISEE-3, the
third International Sun-Earth Explorer. The satellite was languishing in space,
its stock of hydrazine fuel鈥攗sed to adjust its angle and speed鈥攕o
low that any serious manoeuvres were out of the question. This was a great pity,
because comet Giacobini-Zinner was making its way back into the inner Solar
System and ISEE-3 happened to have several instruments on board that would give
useful information about the comet 鈥攊f only the satellite were in the
right place. The engineers needed to move an almost dead satellite millions of
kilometres through space using hardly any fuel.
They did this by exploiting the 鈥渂utterfly effect鈥� from chaos theory, in
which a tiny perturbation鈥攖he flap of a butterfly鈥檚 wings, say鈥攃ould
cause a mighty hurricane. This effect is usually seen as a problem, but in fact
it鈥檚 an opportunity. If the butterfly can be persuaded to make the right flap, a
small effort can give you the large response you desire.
In practice you can鈥檛 do that for weather, because there are billions of
butterflies and other competing influences. But for controlling a satellite, you
can use the effect to advantage because now there鈥檚 no unwanted competition.
Either the engineers expel a small amount of hydrazine fuel from ISEE-3, or they
don鈥檛. The only questions are: what difference does it make, and where should
you do it? That鈥檚 where chaos theory can help.
It鈥檚 all down to the strange way gravity behaves when three or more objects
are involved. Two objects always move around each other in ellipses (or
parabolas or hyperbolas, if the speed is high enough), but three bodies can go
all over the place鈥攁nd tiny disturbances can have huge effects. The Earth,
the Moon and ISEE-3 are three bodies, so chaos theory tells us that if the
satellite is given the right nudge when its orbit is unusually sensitive to
small changes, the resulting effect could be big. And the right moment to do
this is when the satellite is close to the 鈥渘eutral point鈥� between Moon and
Earth, where the Earth鈥檚 gravity exactly cancels out that of the Moon.
So NASA鈥檚 engineers flew the satellite past that point five times, emitting
carefully calculated but very tiny squirts of precious hydrazine on each flyby,
and persuaded the nearly defunct ISEE-3 to become the vibrant and exciting ICE,
the International Cometary Explorer. The spacecraft successfully encountered
Giacobini-Zinner, made the first ever direct measurements of a comet鈥檚 tail, and
confirmed the theory that comets are simply dirty snowballs. Not bad,
considering.
This technique is now known as chaotic control, and it was put on a firm
mathematical footing in 1990 by Edward Ott, Celso Grebogi and Jim Yorke, all at
the University of Maryland at College Park. They devised a systematic
technique鈥攖he OGY method鈥攖o make the butterfly work in your favour,
by repeatedly tweaking a chaotic system in the right manner. The latest idea to
emerge from OGY is a cheap way to send payloads from low Earth orbit to the
Moon. The classical solution is called a Hohmann ellipse. At one end the ellipse
passes close to the Earth, at the other end it grazes the Moon. The theory is
that if you start with a craft in a circular low Earth orbit and use a burst of
fuel to push it onto the path of a Hohmann ellipse, then all you need to do at
the other end is to use another burst to slow it down and switch it into low
lunar orbit.
A Hohmann ellipse may be neat, but unfortunately it鈥檚 not fuel-efficient. So
in 1996 Ott and his co-workers worked out a different orbit using much less
fuel, which makes repeated chaotic swings near the Earth-Moon neutral point. One
snag is that it would take about 10 000 years to follow that orbit. But with
some careful squirts of fuel at crucial places the duration can be shortened to
two years. The resulting trajectory involves 48 irregular circuits of the Earth
followed by 10 circuits of the Moon, after which the payload settles neatly into
parking orbit. It鈥檚 not a hot piece of real estate yet, because we don鈥檛 have
any lunar colonies鈥攂ut if we ever do, the new orbit would convey 83 per
cent more goods to the colony than a Hohmann ellipse, for the same fuel. As long
as the goods aren鈥檛 perishable.
An important extension of the idea of a gravitational neutral point is the
set of Lagrange points discovered by the great French 18th-century mathematician
Joseph-Louis Lagrange. In a system with two orbiting objects where one is much
more massive than the other鈥攖he Sun and Earth, say鈥攖he light body
will go round the heavy one in an ellipse. If you ignore effects of speed, there
will be a single neutral point between the bodies where their gravitational
forces cancel each other out. But there鈥檚 also the accelerating 鈥渃entrifugal鈥�
force associated with moving in a circle. Lagrange discovered five neutral
points where all three forces鈥攇ravity from the heavy body, gravity from
the light one, and 鈥渃entrifugal鈥� force鈥攃ancel. They are called Lagrange
points, L1, L2, L3, L4 and L5
(see Diagram).
In principle a satellite or probe placed at a Lagrange point will stay there
forever. In practice, though, tiny disturbances may cause it to move. If the
heavy body is sufficiently massive compared with the light one, then L4 and L5
are stable positions, but L1, L2 and L3 are always unstable鈥攁 probe placed
there will gradually wander away. However, the whole point of control
engineering is to stabilise the unstable, and it turns out that regular, but
relatively small, expenditures of fuel can keep a probe close to an unstable
Lagrange point for decades.
Cool properties
Several recent and projected space missions are making good use of Lagrange
points. For instance the billion-dollar Solar and Heliospheric Observatory, a
joint project between NASA and the European Space Agency, is hovering near the
L1 point, about 1.6 million kilometres from Earth in the direction of the
Sun鈥攊deal for observing all sorts of solar activity while remaining in
touch with the ground.
The successor to the Hubble Space Telescope will also lurk at a Sun-Earth
Lagrange point, this time L2. The Next Generation Space Telescope (NGST) will
orbit the Sun about 1.6 million kilometres further out than Earth and will
remain in much the same relative position throughout its working life. Hubble,
which was launched in 1990, orbits the Earth, and its structure includes a heavy
tube which keeps out unwanted light that would interfere with the images it
produces. It鈥檚 a lot darker out near L2, so that cumbersome tube can be
dispensed with for NGST, saving launch fuel. In addition, L2 is a lot colder
than low Earth orbit, and that makes infra-red telescopy much more effective.
L2, then, is not so much a hot property as a real cool one.
The prize for the weirdest orbit of all, though, has to go to the planned
Genesis mission, which will collect samples of the solar wind鈥攁 vast
stream of charged particles emitted by the Sun鈥攁nd return them to Earth.
NASA鈥檚 Jet Propulsion Laboratory plans to launch the probe in January 2001. It
must hover in space for long enough to collect samples of the solar wind and
bring them back to low Earth orbit. They will then be returned to the ground in
a capsule with a parachute.
This is a demanding task requiring a complex orbit, but the budget of
$216 million is relatively low, so the more obvious, fuel-hungry
trajectories are ruled out. The chosen trajectory is based on the work of a team
including mathematician Jerrold Marsden and computer scientist Wang-Sang Koon at
Caltech, and Martin Lo at JPL. Genesis will wander near the Sun-Earth L1 point,
collecting its samples and making other observations of the solar wind, and then
return via the L2 point before sinking back into parking orbit by way of a few
close encounters with the Moon (see Diagram).
But why must the spacecraft divert to L2 rather than returning directly to
Earth? It turns out that sending Genesis directly home from L1, inside Earth鈥檚
orbit around the Sun, takes more fuel than sending it back from L2, which is
outside. But to take advantage of this, what you really need is a free detour
from L1 to L2.
That鈥檚 just what Marsden and colleagues found. They discovered a
鈥渉eteroclinic connection鈥濃�攁 constant-energy path linking L1 to L2. It
works like this. Imagine two neighbouring hills, of equal height. The
equilibrium points are the hill tops. Place a ball on top of one hill, and roll
it down into the valley in just the right manner for it to climb to the top of
the second hill. That鈥檚 an example of a heteroclinic connection. In the absence
of friction, the ball moves from one hill to the other using only gravity, which
costs nothing since it is present anyway. Along the way, gravitational potential
energy is converted into kinetic energy as the ball accelerates downhill, and is
then converted back again as it climbs the far hill鈥攂ut the total energy
never changes.
To find exactly what path the spacecraft needed to take to exploit this
heteroclinic connection between L1 and L2, Marsden and his colleagues had to
look for something called an unstable manifold. Remember the ball balancing on
the top of the hill. Give it a push and all directions are down鈥攖hat鈥檚 an
unstable manifold. But now imagine a ball balancing on a mountain
pass鈥攕ome directions are up, or stable, and some down, or unstable. In an
ordinary landscape like this, it鈥檚 easy to spot the unstable direction. But
finding it in space is much harder. For Genesis, the trick has to be played in
an 18-dimensional landscape鈥攕ix coordinates each for Genesis, Earth and
Sun. The unstable manifold now is not a single curve, but a highly complex
multidimensional surface. So until recently, calculating its shape and position
was virtually impossible, taking far too much computer time.
However, in 1996 Michael Dellnitz of Hamburg University developed efficient
mathematical techniques for performing such calculations (see 鈥淕et down鈥�), and
this was the breakthrough needed to find the unstable manifold of the Genesis
orbit. Since it is a multidimensional surface, many different trajectories can
use it to leave L1: the one proposed for the Genesis mission is chosen to fly
past L2 in a manner that makes the final return to Earth especially
straightforward.
Sounds complicated? Not really. With today鈥檚 mathematical techniques it isn鈥檛
much harder to find such an orbit than it is for a prospective purchaser to spot
a seaside cottage with a really nice view. You just need to know what鈥檚
important and how to find it. When you know how to locate those crucial but
invisible features of the celestial landscape, you can be confident of picking
up the best bargains on the sky street.
THE key to computing clever orbits is to find the right direction to enable
spacecraft to make optimum use of the gravitational hills and valleys that they
encounter. Imagine a ball sitting on a mountain pass. Try pushing it up towards
the mountain top and it will simply roll back to where it started. But tip it
over the edge and it will roll downhill. This downhill path is called an
鈥渦nstable manifold鈥� and the question is how to find it.
Michael Dellnitz of Hamburg University has found a way and it works like
this. The tricky first portion of the unstable manifold near the top of the
pass, which is the hard bit to find, is 鈥渁lmost invariant鈥�. This means that the
rolling ball hangs around near it for quite a long time. So one way to find it
is to pick out any orbits that have this characteristic. The problem is that
other orbits have this characteristic too鈥攑eriodic ones for instance. But
it turns out that if you collect together all the nearly invariant ones, it鈥檚
not too hard to pick out the unstable manifold from the rest.
To find the almost invariant orbits, Dellnitz divides the whole
multidimensional space into a coarse grid of cubes. Pick one such cube and
follow all the trajectories that pass through it: make a list of all the cubes
they hit. Now repeat for all the other cubes. From such data, you can calculate
a 鈥渕easure鈥濃�攁 number that tells you roughly how much time the spacecraft
spends in a given cube. If the cube contains part of an almost invariant
pathway, then the spacecraft hangs around nearby for quite a while, and its
measure is big. So you throw away all the cubes with a small measure, take the
ones that remain, and subdivide them into smaller cubes. Now repeat the process,
over and over again. At each stage the cubes that are left provide a better and
better approximation to those almost invariant sets.
If you didn鈥檛 throw most of the cubes away, then the number of cubes you鈥檇
have to track would grow very rapidly鈥攁 thousand at the first stage, say,
then a million, then a billion, then a trillion鈥ife is too short. But because
you throw away most of the cubes at each step, the computation time required
grows much more slowly. And before the calculations become intractable, you can
pin down the almost invariant paths very closely indeed. After that, it鈥檚 easy
to spot the unstable manifold.