Math Games

Manifolds in the Genesis mission

Ed Pegg Jr., September 7, 2004

At 12:15 p.m. EDT on September 8th, 2004, a team of helicopter stunt pilots will catch a paragliding package filled with fragments from the Sun. This will mark the first time NASA has recovered material from beyond the moon's orbit. You can watch the live webcast at Once the capsule has been secured, pilots Cliff Flemming and Dan Rudert will return to Chicago to do stuntwork for the movie Batman Begins.

Leading up to that, on January 21, 1889, King Oscar II of Norway and Sweden celebrated his 60th birthday. It was a large affair -- his father Oscar I died 4 days after his own 60th birthday -- as part of the festivities Oscar II offered a reward for the best scientific paper. One of the challenge questions: Prove that the solar system is stable under Newton's equations. It was a version of the famous three body problem of gravitation. With two gravitational bodies, such as the Earth and Moon, the laws of Kepler and Newton followed easily -- ellipses explained their motion. Add a third body, and the problem became much more complicated.

Henri Poincaré entered the contest with his own analysis, and won the top prize. One of the judges, Karl Weierstrass, described the work to be "of such importance that its publication will inaugurate a new era in the history of celestial mechanics." Poincaré tackled nine simultaneous differential equations, and showed that a solution would have a tendency to converge, using the concepts of homoclinic points, chaotic motion, and invariant integrals. He used the concepts of manifolds to simplify the equations. Unfortunately, as his paper was printing for Acta Mathematica, Poincaré found a major error. He wrote to the editor, Gösta Mittag-Leffler, and asked for his paper to not be published. Poincaré paid for the printing costs, which wound up being substantially more than the prize money Oscar II gave him. After fifty letters back and forth between Poincaré and Mittag-Leffler, a revised version of the paper was published in 1890. The error Poincaré made led to the foundation of Chaos science. (In 1903, he said "small differences in the initial conditions produce very great ones in the final phenomena.")

As mentioned, Poincaré arrived at a solution of the 3 body problem by using "manifolds" - smooth surfaces like an infinite sheet of paper, or the surface of a torus. He showed that some orbits, such as 2-body orbits, became invariant. They remained trapped in a particular manifold, unless something from the outside knocked them out of it. They were like tubes, and the set of all variations stayed on the tube. If an unstable orbit ever returned to its starting point without a loss of energy, Poincaré showed that the orbit stayed forever trapped within these tubelike manifold in six-dimensional phase space. Part of his proof involved the certain points of gravitational stability, first calculated by Joseph-Louis Lagrange. All the major manifolds connect through L1 and L2.

Figure 1. Lagrange Points. Not to scale! (See derivation)

Objects placed at Lagrange points L1, L2, and L3 eventually drift away -- they are repelling equilibrium points. L4 and L5 are attracting equilibrium points -- objects can be found at Jupiter's L4 and L5 points. For Earth, there is a high concentration of dust at both of these points.

L1 and L2 might not be stable, but an orbit around them can be. Eighty years after Poincaré predicted manifolds, Robert Farquhar used manifolds and computers to find a path leading to what he called a "halo orbit". Farquhar persuaded NASA to send the International Sun-Earth Explorer 3 (ISEE3) into a halo orbit around L1 in 1978. There it stayed for awhile, but the manifold tubes of Poincaré extend in many different directions. As celestial objects flit about, the tubes writhe like possessed water sprinklers. Scientists calculated one of these journeys that ISEE3 could take once the original mission completed. In 1985, it was sent to L2, and then through the tail of the Giacobini-Zinner comet. In 1986, ISEE3 was sent close to Halley's comet, then into an orbit around the sun. ISEE3 will return to Earth in 2014, making it the most traveled object ever recovered.

Figure 2. The travels of ISEE3. (Larger image at NASA).

ISEE3 took a journey on what is now called the Interplanetary Superhighway. The manifold tubes of Poincaré are constantly shifting around and intersecting as the planets move. In fact, quite a few objects in the solar system travel on the interplanetary superhighway already -- Comet Oterma has orbited both inside and outside of Jupiter's orbit, with transfers via L1/L2 in 1937 and 1963. Very little energy is needed to travel from tube to tube. It's actually difficult, energy-wise, to avoid these tubes. With patience, one can start from L1 or L2 and go anywhere in the solar system. ISEE3 has richly demonstrated the possibilities of low-energy manifold travel.

In 1999, Wang Sang Koon, Martin W. Lo, Jerrold E. Marsden, and Shane D. Ross published "The Genesis Trajectory and Heteroclinic Connections." Here's the abstract: "Genesis will be NASA's first robotic sample return mission. The purpose of this mission is to collect solar wind samples for two years in an L1 halo orbit and return them to the Utah Test and Training Range (UTTR) for mid-air retrieval by helicopters. To do this, the Genesis spacecraft makes an excursion into the region around L2. This transfer between L1 and L2 requires no deterministic maneuvers and is provided by the existence of heteroclinic cycles defined below. The Genesis trajectory was designed with the knowledge of the conjectured existence of these heteroclinic cycles. We now have provided the first systematic, semi-analytic construction of such cycles. The heteroclinic cycle provides several interesting applications for future missions. First, it provides a rapid low-energy dynamical channel between L1 and L2 such as used by the Genesis Discovery Mission. Second, it provides a dynamical mechanism for the temporary capture of objects around a planet without propulsion. Third, interactions with the Moon. Here we speak of the interactions of the Sun-Earth Lagrange point dynamics with the Earth-Moon Lagrange point dynamics. We motivate the discussion using Jupiter comet orbits as examples. By studying the natural dynamics of the Solar System, we enhance current and future space mission design."

Here is a picture of the route they proposed:

Figure 3. The Genesis Trajectory (from The Genesis Trajectory and Heteroclinic Connections).

The low energy method proposed by Lo and his colleagues was picked over higher-energy missions, and work got underway to prepare a spacecraft that could capture and store particles of solar wind. Part of this involved making five collector arrays, each with 55 hexagonal wafers measuring 10 centimeters in diameter. These wafers consist of 15 different high-purity materials including aluminum, sapphire, silicon, germanium, gold, platinum, and diamond-like amorphous carbon -- all chosen for their durability, purity, cleanliness, retentiveness and ease of analysis.

Figure 4. A Genesis collector array. (Image from NASA/JPL. (Not Dr. Gordon Freeman)).

On April 8, 2001, the spacecraft was launched. For the past 3 years, everything has worked exactly as expected. On November 16th, it successfully started the halo orbit, and the collector arrays were opened. In April 2004, the collectors were closed, and the spacecraft nudged from the L1 orbit to loop around the Earth, then around L2. You can read about the mission at the NASA/JPL site The Story of Genesis.

Back to September 8th, 2004. From Carson City, the 5-foot diameter spacecraft will look like a bright daytime meteor, rapidly traveling from northwest to east across the morning skies. Genesis will hit the Earth's atmosphere at 8:52:46 a.m. PDT over Salem, Oregon. Two minutes later, the capsule will deploy a drogue parachute. Six minutes later, the main parachute, a parafoil, will open up. At 9:15 p.m. PDT, the capsule will be caught by helicopter over Utah. This will be the first NASA collection mission since the Apollo (1972).

Figure 5. September 8, 2004. Helicopter retrieves samples of solar wind. (Image NASA/JPL)

Poincaré would doubtlessly be happy to learn that his methods in manifolds led to the collections of particles from the Sun. Ironically, his other study in manifolds is one of the most important open problems in Mathematics, The Poincaré Conjecture: Every simply connected closed three-manifold is homeomorphic to the three-sphere. Grigori Perelman may have proven this conjecture, hopefully we will know by 2009. At that time, the manifolds of Poincaré will be used again -- to send a mission to the Moon, collect samples, and return them to Earth, on a shoe-string budget! Not a bad outcome for a paper with a major mistake. For up-to-date info on the Genesis mission, see


Neil J. Cornish, The Lagrange Points,

Jet Propulsion Laboratory, "Here Comes the Sun,"

Jet Propulsion Laboratory, "Interplanetary Superhighway Makes Space Travel Simpler,"

Jet Propulsion Laboratory, "The Story of Genesis,"

Wang Sang Koon, Martin W. Lo, Jerrold E. Marsden, and Shane D. Ross, "The Genesis Trajectory and Heteroclinic Connections," (AAS 99-451), Astrodynamics 1999, AAS Vol. 103, Part III, 2327-2343.

Martin W. Lo, Min-Kun J. Chung, "Lunar Sample Return via the Interplanetary Superhighway,"

Martin Lo, "Home Page",

MacTutor History of Mathematics, Joseph-Louis Lagrange.

MacTutor History of Mathematics, Henri Poincaré.

MacTutor History of Mathematics, Karl Weierstrass.

NASA, "ISEE3 Overview,"

NASA, "A Little Glitz Goes a Long Way for NASA's Genesis,"

Ivars Petersen, Newton's Clock: Chaos in the Solar System, W H Freeman & Co, 1993.

Douglas Smith, "Next Exit 0.5 Million Kilometers," Engineering and Science, Volume LXV, Number 4, 2002.

Eric W. Weisstein. "Poincaré Conjecture." From MathWorld--A Wolfram Web Resource.

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Ed Pegg Jr. is the webmaster for He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.