Many folks would like to see us back on the Moon and developing its resources.

Sunday, July 10, 2005

[lunar-update] Capture Dynamics and Chaotic Motions in Celestial Mechanics - With Applications to the Construction of Low Energy Transfers
Good afternoon.

John Reed passed a note.
>From: John Reed
>Sent: Sunday, July 10, 2005 12:46 PM
>To: Larry Kellogg
>Subject: Re: [lunar-update] As noted at Inside KSC - NASA Announces Space
Shuttle Return to Flight Activities

>Greetings--no wind here, in SE Ga, just lots of rain--hope things clear by
Wed, planning to go down a watch the shuttle go off...Some of your
math-oriented readers may enjoy Dr. Belbruno's work:


>By using chaos theory applied to 3-body orbital mechanics, he's found
routes to the moon that are twice as efficient as what NASA used for
Apollo...might be useful when we finally start going there again!


John mentions --- math-oriented readers --- and math was not one of my A
subjects, at least taking Calculus 2, 26 years after Calculus 1 was not the
way to study math with me. (slight break in college with 26 years in the
Navy between my first year and second year of school.)

Still I think you will appreciate that there are different ways of looking
at problems and using your mind to visualize a problem can be productive.
You will see that capture dynamics and chaotic motions in celestial
mechanics paints a picture of sets of orbits that can ease your way from one
point in space to another with little expenditure of energy.

I'll copy the book review below and then some other links and I hope you
will appreciate thinking about going through space on these low energy
transfer routes. Just visualize the roller coaster ride on your space
gravity pin ball machine.

If you read all the way through the last article you may see that a student
of mathematics could spend a great deal of time mapping space for the ride
to the stars.

Now where is my Texas Instruments four function calculator with six
dimension, holographic projection, that folds up in my pocket just waiting
to be used for my next jump point.

Larry Kellogg
larry.kellogg at

[See web site for live links. - LRK -]
Capture Dynamics and Chaotic Motions in Celestial Mechanics:
With Applications to the Construction of Low Energy Transfers

Edward Belbruno

Cloth | 2004 | $52.50 / £33.95 | ISBN: 0-691-09480-2
224 pp. | 6 x 9 | 20 line illus.

Shopping Cart | Reviews | Table of Contents
Chapter 1 [in PDF format]

This book describes a revolutionary new approach to determining low energy
routes for spacecraft and comets by exploiting regions in space where motion
is very sensitive (or chaotic). It also represents an ideal introductory
text to celestial mechanics, dynamical systems, and dynamical astronomy.
Bringing together wide-ranging research by others with his own original
work, much of it new or previously unpublished, Edward Belbruno argues that
regions supporting chaotic motions, termed weak stability boundaries, can be
estimated. Although controversial until quite recently, this method was in
fact first applied in 1991, when Belbruno used a new route developed from
this theory to get a stray Japanese satellite back on course to the moon.
This application provided a major verification of his theory, representing
the first application of chaos to space travel.

Since that time, the theory has been used in other space missions, and NASA
is implementing new applications under Belbruno's direction. The use of
invariant manifolds to find low energy orbits is another method here
addressed. Recent work on estimating weak stability boundaries and related
regions has also given mathematical insight into chaotic motion in the
three-body problem. Belbruno further considers different capture and escape
mechanisms, and resonance transitions.

Providing a rigorous theoretical framework that incorporates both recent
developments such as Aubrey-Mather theory and established fundamentals like
Kolmogorov-Arnold-Moser theory, this book represents an indispensable
resource for graduate students and researchers in the disciplines concerned
as well as practitioners in fields such as aerospace engineering.

Edward Belbruno has been a Visiting Research Collaborator in the Program in
Applied and Computational Mathematics at Princeton University since 1998.
The author of numerous articles in professional journals in mathematics,
astronomy, and aerospace engineering, he received the Laurel Award in 1999
for the salvage of a Hughes satellite in 1998 using lunar transfer.


"You have just launched a multimillion dollar communications satellite into
space. A malfunction means your expensive cargo will not reach its required
orbit. Who are you going to call? A good choice would be Edward Belbruno, a
celestial mechanician who seems to specialize in such lost causes. . . .
Capture Dynamics is the first book to describe the necessary theory in the
context of the whole process of capture. . . . The result is a brilliant
example of the application of chaos theory to practical problems."--Carl
Murray, New Scientist

"This is a how-to book on how to construct a low energy orbit from the earth
to the moon by the man who did just that to save the Japanese space mission
in 1990."--Ken Meyer, SIAM Review


"A classic, by one of the masters of the field. Belbruno found the almost
magical low energy chaotic orbit that got the Hiten spacecraft successfully
to the moon. This text has a beautiful general treatment of chaotic orbits,
which are now of great importance both in the celestial navigation of
spacecraft and in evaluating the threat of killer asteroids in near earth
orbit."--J. Richard Gott, III, Professor of Astrophysics, Princeton

"It is truly amazing to see how many new gems have been discovered recently
in the classical three-body problem, which has led to theoretical delight as
well as to practical applications in solar-system space flight. In this book
we hear the story firsthand, from someone who has been instrumental in both
aspects of these developments."--Piet Hut, Institute for Advanced Study,

"This is a well-written book, the first one in the literature to combine
modern and advanced theoretical results in celestial mechanics with
practical methods used for understanding and designing the low energy
trajectories of space missions."--Florin Diacu, University of Victoria.

[See more reviews and links to table of contents at the web site. - LRK -]

[Often we find that history is made over expanded periods of time rather
than instantaneously. - LRK -]
When The Moon Hits Your Eye ...
By apsmith, Section Commentary
Posted on Sun Jan 11, 2004 at 05:26:55 AM PST
Space Exploration Driving east towards home this evening from the New York
NSS chapter meeting, with my mind on space exploration and all the recent
excitement, I happened to catch the just-past-full Moon exactly on the
horizon, huge and ruddy-brown. Our sister planet, it seemed so close, within
reach. Are we really going back there? And why now?

Dr. Edward Belbruno was the speaker today, and had been intending to talk
about his new book, and how he got into the field of chaotic orbital
dynamics. But first we heard him share some of his excitement about the
potential for the changes (still not yet official) for NASA and US space

Ed just got back from China, in particular Shanghai, where he was stunned by
the vibrancy of the city, the changes of just the last few years, and how
inexpensive everything was (sumptuous dinner for eight people: $12.00). What
really made him both excited for China, and depressed about the current
state of the US, was the repeated sighting on the local news of Yang Liwei,
China's first astronaut, who seems to have caught his country in a frenzy of
space enthusiasm. To Ed it seemed like a combination of New York in the
1900's (when the skyscrapers first went up) and the US in the 50's and
60's - positive, win-win attitudes and enthusiasm and hard work everywhere.

And while he was there, Beagle (is that any name for a spacecraft?)
vanished. Even though China hasn't attempted any lunar or interplanetary
probes yet, it seemed further evidence of the decline of the west... But,
now we have Spirit on Mars, and the promise of a new direction for NASA at
last! Perhaps all is not lost! But some remain skeptical - does NASA really
still have it in it, or is it now like a former Olympic athlete eating
potato chips and drinking beer while watching reruns of past glories.?

Anyway, Belbruno did eventually get to his fascinating talk on chaotic
orbits, capture dynamics, and the story of political machinations at the Jet
Propulsion Lab. Interestingly enough, ESA's Smart-1 craft is following one
of Belbruno's suggested trajectories right now. And Martin Lo's
Interplanetary Superhighway, which was in the news a couple of years back,
is all based on Belbruno's mathematics. Rocket science is actually quite a
bit more complicated than everybody thought!

By the way, NSS has chapters even in remote locations like Alabama; no
admission normally required for meetings :-)

[Notice the date of this IAA symposium and what Dr. BELBRUNO's talk was
about. - LRK - ]
to the Outer Solar System and Beyond

First IAA symposium on realistic near-term advanced scientific space

June 25-27, 1996 - Turin, Italy
International Academy of Astronautics (IAA) together with Politecnico di

Symposium Goals

The Interstellar Space Exploration Committee (ISEC) of the International
Academy of Astronautics (IAA) feels that the time is ripe to exploit many
excellent speculations about space missions to the outer solar system and
beyond up to 1000 Astronomical Units (AU) from the Sun. This Symposium aims
at identifying a Deep Space Transportation System and the relevant space
missions to make new scientific progress, some of which cannot be achieved
within the solar system.

Results from this Symposium will be submitted to the leading Space
Agencies as Proposals for space missions to the outer solar system and
beyond which are feasible within near term technology.

IAA will host a three-day International Symposium on Outer- and
Extra-Solar Missions in the town of Turin (Torino), Italy. Local supporting
organizations will be the "Giuseppe Colombo" Center for Astrodynamics of
Turin and the Departments of Aerospace Engineering and Mechanics of the
Politecnico (Engineering School) of Turin. The Symposium dates will be
Tuesday 25, Wednesday 26 and Thursday 27 June 1996.

Areas of Interest

* Scientific Objectives and Payload Requirements
* Power and Propulsion Systems
* Trajectory Optimization, Mission Profiles
* Attitude Dynamics
* Navigation, Guidance and Control
* Very Deep Space Telecommunications
* Sensors, Detectors and Lens Systems
* Nanotechnology and Robotics
* Perspectives and New Concepts for Missions
* Forms of International Cooperation

BELBRUNO Edward - The Geometry Center USA
GENTA Giancarlo - Dipartimento di Meccanica Politecnico di Torino

Low Energy Comet Rendez-Vous Using Resonance Transitions

[You may want to read the whole article.
See reference to Edward Belbruno towards the end of what I copied. - LRK -]
Next Exit 0.5 Million Kilometers
by Douglas L. Smith

The year is 2020. Under a crescent Earth, the assembly crew at the Lunar
Gateway Service Area some 62,000 kilometers above the moon’s surface
installs new electronics on an infrared telescope and sets it moving, at the
speed of a Ford Pinto climbing an on-ramp, off to park itself deep in Earth’
s shadow, where a little cryogenic coolant goes a long way and where Earth
is always directly overhead for high-speed data downlinks. This spacecraft
has, in fact, just entered what Martin
Lo (BS ’75), a member of the technical staff at JPL, calls the
Interplanetary Superhighway—“a vast network of winding tunnels in space”
that con-nects the sun, the planets, their moons, and a
host of other destinations as well. But unlike the wormholes beloved of
science-fiction writers, these things are real. In fact, they are already
being used.

The Genesis mission, for example, is following a route that a team of
scientists led by Lo plotted through the sun-Earth interchange of this
freeway system. (In fact, this low-energy route helped JPL win the mission.)
Genesis, of which Caltech professor of nuclear geochemistry Don Burnett
is the principal investigator, is collecting samples of the solar wind—the
torrent of charged particles that emanates from the sun, and whose makeup
reflects that of the disk of gas and dust from which the sun and the planets
condensed. In 2004, Genesis will bring its booty home the same way. (Well,
technically not the same way, as it will return through the other side of
the cloverleaf,
as it were.)

Lo, together with Caltech’s Jerrold Marsden, professor of control and
dynamical systems (CDS), and their coworkers Shane Ross (BS ’98, a CDS
graduate student) and Wang Sang Koon (a CDS senior postdoc) have begun a
systematic mapping effort of what is more properly known as the
Interplanetary Transport Network. As a freeway system, the network is more
akin to the Pacific Coast Highway and other scenic routes than to
interstates like the I-5—a collection of meandering byways for leisurely
travel, not the fastest, most direct routes between points. But the quickest
paths in outer space are all toll roads (it costs a lot of rocket fuel to
use them), while you can ride the Interplanetary Superhighway almost for
free. Gravity does the driving, so the system is really more like an
elaborate set of Hot Wheels tracks. All you have to do is let go of the car
at the right place. (It’s a lot more complicated than this, because the
tracks are in constant motion, but we’ll get to that later.)

Think of a planet as a bowling ball sitting on a taut rubber sheet; the
depression the ball makes is its gravitational well. To hoick a marble (the
spacecraft) up out of that well takes thrust—often quite a lot of it. But
what if the marble was balanced on a cusp, such as where Earth’s well and
the moon’s well meet? The gravitational (and sometimes rotational) forces
would balance one another, and the slightest sneeze, a mere feather touch,
would nudge the spacecraft in the right direction.

A set of five of these balance points, called Lagrange or libration points,
exist between every pair of massive bodies—the sun and its planets, the
planets and their moons, and so on. Joseph-Louis Lagrange (1736–1813)
discovered the existence of the two points now known as L4 and L5, each of
which is located in the orbital plane at the third vertex of an equilateral
triangle with, say, Earth at one vertex and the moon at the other. So L4 is
60° in advance of the moon, and L5 60° behind it. Ideally, a spacecraft at
L4 or L5 will remain there indefinitely because when it falls off the cusp,
the Coriolis effect—which makes it hard for you to walk on a moving
merry-go-round—will swirl it into a long-lived orbit around that point.
Comet debris and other space junk tends to collect there, and Jupiter has
accumulated an impressive set of asteroids that way.

Leonhard Euler (1707–1783) rounded out the assortment with L1, L2, and L3,
where the rotational forces don’t balance out and nothing stays put for
long. (Euler actually discovered L1, L2, and L3 first, but Lagrange had a
better press agent.) For the sun-Earth pair, L1 lies on a line between them,
about 1.5 million kilometers from Earth. Genesis is parked in a halo orbit
around L1, so called because, as seen from Earth, the flight path follows a
halo around the sun. (Sitting right on L1 isn’t a good idea, as the
spacecraft’s radio signals would be lost in the sun’s glare.) Since orbits
around L1 are unstable, Genesis needs a small boost every 60 days or so to
keep it on station. Because of its unobstructed view of the sun, L1 is a
popular place these days—the Solar and Heliospheric Observatory (SOHO), a
joint project of the European Space Agency and NASA, and NASA’s WIND and the
Advanced Composition Explorer (ACE) are also there. (Ed Stone, the Morrisroe
Professor of Physics, is ACE’s principal investigator, and two of its nine
instruments were built on campus.) L2 is a similar distance from Earth as
L1, but in the opposite direction, and L2 orbits are also unstable on a
60-day scale. NASA’s MAP, the Microwave Anisotropy Probe, has been in orbit
around L2 since October 2001, mapping the variations in the leftover heat
from the Big Bang. And finally, L3 lies hidden from our view directly behind
the sun. We haven’t found any reason to keep a spacecraft there, but it’s
proven a dandy place for science-fiction writers to conceal inhabited,
Earthlike planets—despite the fact that all these anti-Earths and Planet Xs
would, if left unattended, fall out of orbit in 150 days or so.

“Historically, L1 and L2 were not interesting,” says Koon. “People were
interested in L4 and L5, because they are stable. But instability can be a
good thing, because a little force achieves a big result.” Adds Ross, “You,
walking around, are dynamically unstable. You keep falling forward.” Without
your inner-ear balance system constantly sending signals for your body to
right itself, you’d be flat on your face in an instant. (Watch a toddler
learning to walk some time.) But this imbalance is good, as it allows a
little forward impetus to move your mass.

The tools to deal with this celestial instability were developed by
Jules-Henri Poincaré (1854–1912) in the 1890s. Poincaré was working on the
infamous three-body problem, which has bedeviled mathematicians since the
days of Isaac Newton. The problem is simplicity itself: calculate the orbits
of three masses whose only interaction with one another is through their
gravitational pulls. Building on Kepler’s foundational work, Newton solved
the two-body version—Earth going around the sun, for example—but throwing a
third mass into the mix gives a complex interplay of constantly shifting
forces. Says Marsden, “You can fit the equations for the three-body problem
in a corner of the blackboard somewhere, but the subtleties in it are very
interesting. Many computational scientists like working on it because it’s
one of the simplest problems that’s complicated enough to test out
computational theories.” And trying to do all nine planets at once?

Poincaré simplified the mess by organizing similar orbits into “manifolds.”
A manifold is any nice, smooth surface: a sheet of paper is a manifold, as
is the surface of a sphere; or the crust of a donut, also called a torus.
Poincaré saw that families of orbits lay on “invariant” manifolds—“No matter
where it goes, a particle that starts on that surface will remain on that
surface forever unless you give it a knock,” Lo explains. “So that surface
is invariant.” These manifolds sit inside what is called six-dimensional
phase space, because it includes the three dimensions of normal space plus a
dimension for the particle’s velocity in each direction. Thus particles that
have the same location but different velocities will appear at different
points in the 6-D phase space. (Marsden and Lo are working with Alan Barr,
professor of computer science, on ways to visualize such higher-dimensional
objects, but, fortunately, there’s a lot that’s easy to see in two


In yet another leap, the collaboration includes a pack of chemists—Charles
Jaffé of West Virginia University, Turgay Uzer from Georgia Tech, and Utah
State’s David Farrelly. “Suppose an asteroid hits Mars and throws up a bunch
of debris,” Marsden asks. “What’s the probability of some of it reaching
Earth? Being bound in Mars orbit and then escaping is mathematically
analogous to a molecule breaking apart. Jaffé, Uzer, and Farrelly brought in
all sorts of techniques from chemistry, because chemists have really been
worrying about those problems.”

It’s useful to know how material sloshes through the solar system, drifting
on gravity’s currents, but no man will wait for that tide. A mission a grad
student would consider a good career move needs to get where it’s going in
only a few years at most. You can do this if you stay in the vicinity of one
planet—taking the beltway rather than the inter-state, as it were—and a
Japanese spacecraft named Hiten was the first to do just that. Launched in
January 1990, it was placed into a highly elliptical orbit around Earth,
from which it was supposed to release a small probe named Hagoromo into
lunar orbit. Hagoromo’s radio failed before deployment, however, and Hiten
didn’t have enough fuel to get to the moon itself on a conventional path. So
JPL’s Edward Belbruno (now affiliated with Princeton) and James Miller
proposed gradually nudging Hiten’s orbit into a very long ellipse extending
some 1.4 million kilometers from Earth. (You may recall that the sun-Earth
L1 point is about 1.5 million kilometers out.) In this region, which
Belbruno and Miller called the Weak Stability Boundary, a carefully timed
rocket burst sent Hiten looping into lunar orbit.

The same trick will work in other planetary neighborhoods. The gulf between
Jupiter’s moons Ganymede and Europa is only about 400,000 kilometers—about
the distance from Earth to
the moon. So a mission could fly to Jupiter on conventional conic sections,
then slip into a “petit grand tour” of the Jovian system. Based on the
method developed in the Chaos paper, Koon, Lo, Marsden, and Ross created a
proof-of-concept flight plan in which a spacecraft took one loop around
Ganymede before settling into a permanent orbit around Europa—a moon that
planetary scientists are dying for a long look at, as it may have oceans of
liquid water beneath its frozen surface. “Most previous applications of
dynamical systems theory to mission design focused on the surfaces of the
invariant manifolds,” says Koon. “We showed that the regions inside and
outside the manifolds can be used to advantage as well as the manifolds
themselves.” Recent, more ambitious itineraries include Callisto and Io, and
go on for a hundred or so orbits. Says Ross, “I think 100 is sort of the
limit of predictability. Like you can only predict the weather for so many
days into the future, we can only predict what’s going to happen up to some
finite horizon in any chaotic environment.” So you still need to fire the
maneuvering thrusters every now and then, just as Genesis needs a nudge
every couple of months to keep it in orbit around L1. But you’ll use a lot
less fuel if you work with the manifolds rather than trying to punch through

Nowadays, these calculations for space missions are done using a software
package called LTool, which can actually handle an N-body problem using
positional data from JPL’s elaborate model
of the solar system, called an ephemeris. LTool grew out of a set of
computational tools developed by astrodynamicists at Purdue over the last 20
years, and which Lo borrowed from longtime collaborator Kathleen Howell, a
halo-orbit expert there. Lo, Howell, and her grad student Belinda Marchand
used these tools on a Jupiter-resonant comet called Helin-Roman-Crockett
(codiscovered by JPL’s Eleanor Helin), meticulously matching the comet’s
orbit with the appropriate segments of Jupiter’s tubes—the model’s first use
of the real motions of actual celestial bodies instead of idealized circular
orbits. In 1998, Lo assembled a JPL team to develop and expand the software,
calling on Larry Romans (PhD ’85), George Hockney, Brian Barden and Roby
Wilson (both Howell alums), Min-Kun Chung (BS ’81), and James Evans.


But right now Lo, Marsden, and company are a long way from compiling a Rand
McNally–esque road atlas of the entire system, much less a detailed street
map. The work they’ve done so far is more of a point-to-point nature—the
equivalent of having a bunch of people just hop into cars and drive, and
seeing where everyone winds up. To do this, they pick a large number of
initial conditions and let the computer run the paths out to wherever they
go. They can make informed choices of the starting points to set the machine
off in the right direction, but the final destinations remain the luck of
the draw. What they need is enough computer power to make the equivalent of
time-lapse aerial photographs of all those flailing tubes. Furthermore,
there are whole families of periodic and quasi-periodic orbits that haven’t
even been catalogued yet, much less explored—you can have an orbit that lies
on the surface of a torus, for example, so that it looks as if you had
soldered the ends of a Slinky together. The manifolds winding onto and off
of this orbit look like hairy donuts. And Randy Paffenroth, staff scientist
in applied and computational math; Eusebius Doedel, visiting associate in
applied math; Herb Keller, professor emeritus of applied math; and Don
Dichmann of the Aerospace Corporation have discovered even weirder orbits
that are impossible to describe in simple terms.

But what if you aren’t anywhere near the tube you want to take? Lo, Marsden,
and company, in collaboration with a group headed by Michael Dellnitz at the
University of Paderborn in Germany, are also working on an extension of the
theory they call “lobe dynamics.” Lobe dynamics allows you to begin at a
distant point on the Poincaré cut and, over many successive passes, hop
chaotically between orbital resonances until you arrive in the vicinity of
the proper tube and fall in. It’s kind of like starting in the middle of a
grassy paddock and bouncing your way over the rough ground to reach a paved

“Exactly how three-body dynamics can be used to help solve Galileo-type
trajectories is a real-life research problem,” says Lo. “How this connects
up to conventional conic orbits is not entirely clear. They’re not separate,
not clearly distinct, and we would like to be able to use elements of both.
Ultimately, this will enable us to do missions we can’t even conceive of
now. The rational numbers didn’t replace the integers, they just increased
the number of things we could do. And who knows what lies ahead? Beyond the
rationals are the irrational numbers, and beyond them the imaginaries.” Says
Marsden, “Since the foundation of the Interplanetary Transport Network we
have laid is so broad and fundamental, it helps us understand many
different, otherwise disparate phenomena at multiple scales, from the
trajectories of spacecraft to the chaotic motion of comets and the transport
of zodiacal dust particles.”

Says Marsden, “We’d really like to establish a formal joint Caltech/JPL
center to carry on this research, perhaps in collaboration with Caltech’s
Center for Integrative Multiscale Modeling and Simulation. But we need a
donor—someone who thinks this stuff is really cool.” Adds Lo, “It’s a
large-scale project. Maybe not quite as horrendous as, say, the human
genome, but a lot like those star catalogs. It’s going to take some time,
because there are a lot of theoretical underpinnings that are still not
really understood.” The center would take advantage of Caltech’s
supercomputer facility and draw faculty from a wide range of disciplines.
And the work done at the center would redound
to other fields in return—perhaps the celestial dynamicists will ultimately
teach the chemists
a thing or two.



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