The Interplanetary Transport Network

Some mathematical sophistication allows spacecraft to be maneuvered over large distances using little or no fuel

Shane D. Ross

eople often picture the solar system dynamics have recently revealed the this way is a straightforward exercise. Pas a cosmic clockwork. And why existence of a complex set of allowable The person planning the trajectory not? With few exceptions, the planets trajectories for such objects—an inter- need only to consider the influence of orbit the Sun in near-perfect circles, planetary transport network of criss- one celestial body at a time. That is, and the moons orbit their planets in crossing pathways. These invisible you can treat the departure from Earth the same manner, all moving with the highway lanes, originating near a planet and the arrival at a distant planet each famous regularity of the heavens. One or moon, guide traffic through the solar to be interactions between the space- imagines the gravitational field created system. But unlike the thoroughfares craft and one massive body. Similarly, by this orderly mechanism to be equal- one finds on the ground, the space high- the transfer from Earth’s general neigh- ly regular. Drop a rock or spacecraft ways and their interchanges are dynam- borhood to that of a faraway planet somewhere close to the Sun, and the ic, with lanes moving past one another may be worked out by considering the object should plummet into the huge according to the varying geometrical spacecraft and the Sun alone. Hence solar mass; release it somewhere near relations between planets and moons. one only has to deal with two bodies our planet, and it ought to drift, per- Staggering through this tangled web, at a time. The overall path can then be haps more slowly, back to Earth. comets and asteroids find themselves approximated using an appropriately Nature, alas, is not so simple. The jumping from one lane to another willy- linked series of simple curves: ellipses, underlying complication, of course, is nilly, getting handed off between plan- hyperbolas and parabolas—the two- that the Earth is orbiting the Sun, not ets—or sometimes running into them. body problem’s well-known “conic just hovering fixed in space. As a re- For such pieces of cosmic flotsam, the solutions” (these being the curves one sult, some very unintuitive things can solar system turns out to be more like a gets after slicing a cone with a plane), happen. A rock let go near our planet turbulent sea than a clockwork. which were discovered by Johannes could find itself following a complex Although people cannot influence Kepler in the 16th century. NASA’s and chaotic path, perhaps orbiting the fates of speeding comets or aster- spectacular multiple flyby missions of first the Earth, then the Sun, and back oids (not yet, anyway), mission plan- the outer solar system, such as Voyager again, over and over for years. Add ners do maneuver spacecraft and can 1 and 2, were based on such a patched- in the tugging of all the other planets direct them to jump from lane to lane conic approach, which was used to and moons, and the possible routes on the interplanetary highway in such provide an initial guess for a numeri- through space can get enormously a way that they can travel vast dis- cal procedure that produced a more complicated—and quite interesting. tances using practically no fuel. Like precise solution by taking into account Investigators from fields as diverse an island castaway who throws a mes- all gravitational and nongravitational as mathematics, chemistry and fluid sage-laden bottle into the right cur- influences on the spacecraft. rent at the right time, the controller can For missions sent out to fly past Shane D. Ross earned a Ph.D. in control and dy- send a spacecraft to gravitational sweet multiple bodies—say, Jupiter, Saturn, namical systems from the California Institute of spots that provide natural gateways to Uranus and Neptune in the case of the Technology in 2004. Since that time he has been an more distant destinations. Harnessing Voyager 2—the speed of the spacecraft NSF Mathematical Sciences Postdoctoral Fellow at this effect to good purpose, specialists relative to these planets is high, and the University of Southern California, and in Au- can plan fuel-efficient routes, ones that the time during which the Sun and a gust he will join the faculty of Virginia Polytechnic would not otherwise be imaginable or planet produce comparable accelera- Institute and State University in the Department technically feasible. tions on the spacecraft is very short. So of Engineering Science and Mechanics. Ross’s the patched-conic approach works very research interests include the study of spacecraft Buck Rogering Through Space well. One drawback with this approach, control and mission design, geometrical methods for engineering systems, and mixing and transport Most interplanetary travel doesn’t ex- though, is that planetary flybys end up processes. Address: Department of Aerospace and ploit subtle gravitational effects. In- being very brief. Another is that fuel be- Mechanical Engineering, University of Southern stead, spacecraft race quickly to their comes a major factor limiting the space- California, RRB 217, Los Angeles, CA 90089-1191. destinations Buck Rogers style, using craft’s itinerary. A prohibitively large Internet: www.shaneross.com chemical rockets. Blasting around in amount of propellant would be needed,

© 2006 Sigma Xi, The Scientific Research Society. Reproduction 230 American Scientist, Volume 94 with permission only. Contact [email protected]. Figure 1. Spaceflight typically requires the expenditure of considerable quantities of propellant. But after it blasted off from Earth, the Genesis probe was able to travel 1.5 million kilometers toward the Sun (green portion of the trajectory), which is some four times farther than the Moon’s orbit (gray circle). Genesis then orbited the Earth’s L1 Lagrange point (white cross in foreground) collecting particles of the solar wind for two and a half years before traveling millions of kilometers along a circuitous path (blue) that looped by another Lagrange point, L2 (second white cross), before returning to Earth in September 2004. Amazingly, Genesis completed this vast trek using hardly any fuel. The probe did so by following one of the many possible low-energy paths through the solar system, routes that have long served as natural conduits between planets for asteroids and comets. Some of these conduits lead to collision with Earth, as the Genesis probe’s path did by design. for example, to put on the brakes and back to Earth in 2004. In an unfortunate mount such a journey through space insert a craft into orbit around some dis- mishap, the parachute failed to deploy is similar to what sailors have long tant planet or moon, observe for a while after re-entry, and the sample-return done—taking advantage of ocean cur- and then blast off to the next destina- canister was badly damaged when it rents to speed them where they want to tion. And taking extra fuel for maneu- struck the ground at high speed. Thank- go. Ancient mariners often discovered vering means that the scientific payload fully, scientists were able salvage some natural currents by noting the motion must be made smaller than would oth- of what was collected, the first extrater- of driftwood or seaweed being carried erwise be possible. Thus mission plan- restrial material brought back to Earth with them. To some extent modern ners have to strike a balance between from deep space since the last of the space navigators can do the same, ob- the proposed trajectory and the amount Apollo landings in 1972 and the first to serving the movement of natural ob- of instrumentation that can be carried. be collected from beyond the Moon’s jects, namely comets and asteroids. The Jupiter-bound Galileo probe and orbit. Genesis completed its journey of A comet called Oterma is particu- the Apollo lunar lander, for example, more than 30 million kilometers using larly interesting in this regard. Ear- began their journeys away from Earth only a minimal amount of propellant. ly in the 20th century, this icy body with about half of their masses being For comparison: A car with a full tank circled the Sun outside Jupiter’s or- made up of fuel. of gas (also about 5 percent of the vehi- bit. Then, after passing close to that In another category entirely was the cle’s total mass) can go only about 500 planet in 1937, Oterma began to orbit Genesis Discovery Mission to sample kilometers before it’s time for a refill. inside Jupiter. The two bodies met the solar wind, which used only 5 per- Voyages like that of the Genesis up again in 1963, at which point the cent of its mass for fuel. Launched in spacecraft would have been inconceiv- comet moved back to the outside of 2001, the Genesis spacecraft flew 1.5 able not long ago, but they are now Jupiter, where it remains today. Dur- million kilometers toward the Sun, possible thanks to the better appre- ing each of its encounters with Ju- where it loitered for two and a half ciation of the low-energy passageways piter, the comet loosely orbited the years gathering individual atoms of the that wind between planets and moons. planet. That is, for a time Oterma solar wind, ultimately bringing them Conceptually, the approach needed to was a captured moon.

© 2006 Sigma Xi, The Scientific Research Society. Reproduction www.americanscientist.org 2006 May–June 231 with permission only. Contact [email protected]. �� Tubes and Funnels �� Building on Kepler’s work of the pre- ��

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� � each other. Allowing that one of the three bodies is much smaller than the �� other two helps to make the problem �� more tractable, but it still remains a thorny one to solve. Many scientists have worked hard on it through the years. Newton tried and got fed up. Figure 2. Mission planners have come to appreciate that in certain cases the best routes for He wanted to write down an equation spacecraft are not always the most direct ones. In some instances it may be smarter to take advantage of the low-energy pathways connecting key points in space. For example, a spacecraft that describes the motion of the third body for all time. But he failed, and destined for the surface of the Moon might get there via one of the lunar L1 or L2 Lagrange points (green path). Such Lagrange points may also serve as way stations for trips to other the three-body problem was declared planets, as shown in this space “subway map.” (Courtesy of NASA.) unsolvable. Let’s not give up so easily. The trick What made this comet move along situation is referred to as the restricted is to not seek a tidy little equation to such a strange path? The best way to three-body problem (restricted by the describe the motion. Instead, one can get a sense of the answer is to simplify requirement that the third body have proceed by considering the problem the problem and consider the motion negligible mass compared with the from a geometrical point of view, seek- of the comet (a relatively small object) other two). Although the full solu- ing intuitive insight into what the so- as it is being acted on by the gravi- tion to this problem is rather hard lutions might look like. For this, it is tational tug of two massive bodies: to fathom, the key elements can be helpful to think of one of those funnel- in this case, the Sun and Jupiter. In understood with just a little physical shaped contraptions one sees every the study of celestial mechanics, this insight. now and then at science museums, often called “a gravity well,” by virtue of the physical analogy with gravitation. A little chute on the side allows you to roll a coin into the device in such a way that it will roll around the inside of the funnel for quite some time. A marble would travel in the same way. In a frictionless world, a coin or marble would just keep circling around such a funnel, mimicking the orbit of, say, Earth around the Sun (if one as- sumes that the funnel is standing in for the Sun’s gravitational well). But in the real world, a circling coin or marble suffers some frictional loss. So over time its orbit around the funnel decays, causing the coin or marble to spiral inward and downward. This tendency is easily observed and indeed forms the economic basis for many of these devices: They eventually take your money. But in watching your change disappear, you’ll learn some things. In particular, you’ll notice that the small-

Jimmy Dorff Dorff Jimmy er the size of the orbit, the more times Figure 3. The existence of low-energy passageways through space can be understood on an the coin goes around in a given pe- intuitive level by considering the physics displayed by a “gravity well.” These funnel-shaped riod. That is to say, smaller orbits have devices (like the one shown here, located at the Morehead Planetarium in Chapel Hill, North higher angular frequencies. The same Carolina) allow coins to circle stably much the same way that a planet orbits the Sun. is true for objects orbiting in space.

© 2006 Sigma Xi, The Scientific Research Society. Reproduction 232 American Scientist, Volume 94 with permission only. Contact [email protected]. Figure 4. Marbles circling in a gravity well—or planets circling the Sun—do so at a rate that depends on the size of their orbits: The smaller the orbit, the larger the angular frequency required to achieve balance (left). A demonstration device engineered to mimic the gravitational field that arises from a pair of massive objects, say the Sun and Earth, would be shaped like a large funnel with a small funnel embedded in it (lower right). Here the small funnel would have to orbit the large one, just as Earth orbits the Sun. A marble traveling around the large funnel at the same angular frequency could balance at two spots that straddle the small funnel (white crosses beneath marbles)—corresponding to Earth’s

L1 and L2 Lagrange points. With care, a marble could be positioned and given an initial velocity such that it would then “orbit” such special locations (at least for a limited time), just as Genesis was made to orbit Earth’s L1 Lagrange point (upper right).

Imagine now a funnel with three has a small funnel shape embedded in moving too slowly to support itself marbles circling in three closely spaced, the main funnel. What is more, let that against the steep walls of the main fun- parallel orbits, one just a bit farther out small funnel circle around the larger nel and would be drawn inward. The than the next. Compared with the one one at the same rate that a coin or mar- only place a marble with that particu- in the middle, the marble in the outer ble would have moved around at that lar angular frequency circles properly orbit would have to travel at a slower position. This arrangement mimics the is at the radius of the Earth-funnel’s angular frequency to be stable; the one gravitational wells of, say, the Earth orbit—or is it? in the inner orbit would go around at (the small funnel) and Sun (the large Closer scrutiny of this weird surface a higher angular frequency. The same funnel) combined. will reveal two very special points. is true in space, say, for three aster- Consider now what would happen One lies close to the ridge that con- oids orbiting the Sun. If the middle one if you shot a marble around in such a nects the little Earth-funnel to the main were traveling at one astronomical unit way that it circled the central depres- Sun-funnel. Let’s get there by start- from the Sun (150 million kilometers, sion at the same distance from the cen- ing near the center (deep in the Sun- the Earth-Sun distance), it would take ter and at the same rate as the small funnel) and moving in the direction 365 days to make one revolution. The (Earth) funnel—but not too close by. of the Earth-funnel. The surface first outer asteroid would take slightly lon- The marble would just circle around rises with its usual steepness, but then ger than 365 days to make a full circle; nicely for a long while. If it were mov- it rolls over—that is, the slope first the inner asteroid would complete its ing at the same angular velocity but becomes less steep, then things flat- orbit faster than 365 days. positioned farther out (where the ten out, then you drop into the Earth- This pattern is easy enough to un- surface has a gentler slope), this mar- funnel. Before getting to the top of the derstand when you recall what hap- ble would be going too fast to circle intervening ridge, you’ll pass a point pens with one of those coin-and- around stably and would be flung out- where the slope is just right for balanc- funnel gizmos. But now imagine that ward. Conversely, if it were inside of ing a marble that is circling around one of those science-museum devices the Earth-funnel’s orbit, it would be at the same rate as the Earth-funnel.

© 2006 Sigma Xi, The Scientific Research Society. Reproduction www.americanscientist.org 2006 May–June 233 with permission only. Contact [email protected]. that on this side, a marble would nor- derstanding the interplanetary trans- mally orbit at an angular frequency port network, because they form key �� that is less than that of the Earth-fun- gateways to faraway destinations.

nel. A marble that moved with the Although L1 and L2 are classified angular frequency of the Earth-funnel as unstable points, that categorization �� but positioned farther out, where the can be misleading, because spacecraft walls of the main funnel slope less can stick around these points for long �� steeply, would normally be expected periods of time. Indeed, the delicate �� to fly outward. But there is one spot interplay of gravitational and rotation- where it won’t do that: just on the al forces allows a spacecraft to move

outside of the Earth-funnel, where about these points, “orbiting” L1 or the surface is somewhat steeper than L2 in the rotating frame of reference, �� normal. Again, there is an equivalent even though there is no material object �� balance point in space, located 1.5 there. Although such orbits around a million kilometers from Earth in the mere point in space appear very bi- Figure 5. Five points of gravitational equi- direction opposite the Sun. zarre, they are, in fact, nothing more librium exist in the restricted three-body The 18th-century mathematician than near misses to being exactly on problem, as shown here for the Earth-Moon Leonhard Euler discovered these two L1 or L2 and moving at just the right system. All five move with the Moon as it special points (along with a third). His velocity for perfect balance. orbits Earth. Lagrange points L , L and L 1 2 3 contemporary Joseph-Louis Lagrange To understand better, imagine that are considered unstable, because an object discovered two others, and the five a space probe was orbiting around the placed at one of these three points will tend to drift slowly away from it over time. The are now known as Lagrange points. Sun close to Earth’s L2 point but just Although each represents a special a little bit to the inside of it. Assume L4 and L5 Lagrange points are stable in the sense that objects placed in their vicinity will orbit around the Sun, they are called too that it was moving just a little bit naturally tend to remain close by. “points” because they appear as fixed faster than it needed to go had it been

locations when viewed in a reference positioned right on L2. What’s it go- Remember, you’d normally expect this frame that rotates at the same rate that ing to do? Again, visualizing a marble marble to have to circle around faster the Earth and Sun orbit around their circling around in a double-funnel ar- to stay balanced. But because the slope center of mass (a point deep inside rangement helps. A marble with the here is somewhat less steeply inclined the Sun). Five such special spots, des- corresponding position and veloc- than is typically the case for this orbital ignated L1 through L5, exist for every ity to this space probe would start to radius, the marble can circle around pair of massive bodies—the Sun and a move ahead in its orbit around the just fine. This point of balance for the planet, a planet and one of its moons, Sun-funnel (ahead compared to L2); it marble has an equivalent in space. It’s and so on. L1 corresponds to the inner would also tend to be flung outward located 1.5 million kilometers from balance point for the marble described slightly. But the surface around here Earth in the direction of the Sun. above (the one located between the has a strange shape. So as the marble

Getting back to the funnel realm, on Earth-funnel and Sun-funnel); L2 cor- moves slightly ahead and outward, the outside of the Earth-funnel there is responds to the outer balance point. the surface in front of it rises, causing another special balance point. Recall L1 and L2 are of direct interest for un- the marble to slow. Soon L2 catches up with it (on the inside). The marble then

begins to trail L2 and encounters another rising surface behind, which acts to speed the marble up and to scoot it �� toward the Sun, just as a wave propels �� a surfer toward the beach (and often a little sideways). So the marble ends �� up pretty much where it began and �� with about the same velocity. It might

successfully “orbit” L2 a few times in this manner before either being flung outward or falling off into the nearby Earth-funnel.

A spacecraft positioned near L1 can act similarly. Viewed from the perspec- �� tive of someone on Earth, the craft

would appear to orbit Earth’s L1 La- grange point for a while and then go Figure 6. A spacecraft given the proper initial velocity can be sent along a trajectory that would shooting off toward Earth or around the Sun—all without expending any then carry it into orbit around, for example, Earth’s L2 Lagrange point (pale green line). A collection of similar trajectories (any point on which the spacecraft would have a specific position fuel. Some of the possible orbits about and velocity) constitutes one “tube” of the interplanetary transport network (green mesh). A these two Lagrange points lie in the spacecraft on a trajectory inside this tube will pass L2 and head toward the outer solar system plane of Earth’s orbit. Others, like the (blue line), whereas one on a trajectory to the outside will fly back toward the Sun (red line). one followed by the Genesis probe, are

© 2006 Sigma Xi, The Scientific Research Society. Reproduction 234 American Scientist, Volume 94 with permission only. Contact [email protected]. three-dimensional and have a variety of spiraling shapes, dipping into and out of the orbital plane of the two massive bodies.

Surfing Between Planets At the end of the 19th century, the French mathematician Henri Poin- �� caré made significant strides in understanding of celestial mechanics at work here. Poincaré was the first to appreciate the complicated motion of the third body that could result. The geometric methods he used to come to this conclusion laid the foundation for what is now known as nonlin- Figure 7. Some tubes of the interplanetary transport network lead objects into orbit around

ear dynamics, more generally called Earth’s L1 and L2 Lagrange points (green trajectories, right), whereas others lead objects away chaos theory. It is important to keep from such orbits (orange). Mission planners can make use of the intersection of these incom- in mind that “chaotic” does not mean ing and outgoing tubes, directing a spacecraft to hop from one tube to another in a way that allows it to travel between L and L or into orbits around the Sun that can be smaller or random. Chaotic-looking paths exist in 1 2 this problem, but they are nevertheless larger than that of Earth (left). However, a spacecraft with limited kinetic energy (say, just the amount needed to orbit L or L ) cannot visit certain regions (gray) no matter what tube predictable, at least for a while into 1 2 pathway it then follows. the future. So a mission designer with

sufficient understanding can take full that shifts from an orbit that is inside a cinity of L1 or L2) and wind around advantage of them to work out vari- planet’s orbit to an orbit that lies out- whatever two massive bodies are be- ous low-energy routes through space. side must pass along them. Like water ing considered, stretching and twisting Poincaré brought order to the chaos directed by a hose, the set of possible along the way. One can think of there

by organizing similar paths into spe- planet-passing objects is imagined being a gateway region around L1 and cial collections of surfaces, which ex- to flow along these tubes, but in six another around L2, with the tubes be- ist in what mathematicians refer to as dimensions instead of just three. The ing the passageways in and out of the a “six-dimensional phase space,” one comparison with fluids is more than domain of the planet or moon. Anoth- that includes the three dimensions of just analogy. Indeed, computational er property of objects traveling along normal space (say, x, y and z) and three tools originally designed by Francois such a tube is that they will move at dimensions for an object’s velocity in Lekien of Princeton University and his their slowest relative to the nearby each direction. co-workers for computing dynamical planet or moon when in the gateway, Building on Poincaré’s work, in the channels in the ocean have been used which can be thought of as a region of late 1960s Charles C. Conley (a math- to ascertain low-energy trajectories in near-equilibrium, the top of an ener- ematician then at the University of Wis- the celestial context as well. getic hill that objects must climb and consin) discovered a collection of tube- Computing the configuration of overcome. shaped surfaces for objects under the these tubes out farther than Conley It turns out that Oterma’s strange gravitational influence of two mutually or McGehee were able to do, my col- path lies along the tubular passage-

orbiting bodies, a result later pursued leagues and I found that they extend ways connected to Jupiter’s L1 and L2 by Robert P. McGehee, then Conley’s far from their region of origin (the vi- Lagrange points—almost as if the com- student and now at the University of Minnesota. An object located on one of these six-dimensional tubes (which is to �� say, having just the right position and �� velocity) will naturally be carried toward or away from a trajectory that or- �� bits about the L or L Lagrange points 1 2 �� as seen in the rotating frame of refer- �� ence. Trajectories on the inside of such �� a tube snake past the Lagrange point, �� whereas those on the outside end up with the object bouncing back. �� Starting in the mid-1990s, I have �� worked with Martin W. Lo of NASA’s Jet Propulsion Laboratory (JPL) and �� Wang Sang Koon and Jerrold E. Mars- den of the California Institute of Tech- Figure 8. Comet Oterma followed the interplanetary transport network from an orbit that was nology to extend this approach. We’ve outside Jupiter’s in 1910 to an orbit that was inside Jupiter’s between 1937 and 1963, when it shown that the important physical once again shifted to an outside track. Its curious route through space is shown here in both property of these tubes is that anything fixed (left) and rotating (right) reference frames.

© 2006 Sigma Xi, The Scientific Research Society. Reproduction www.americanscientist.org 2006 May–June 235 with permission only. Contact [email protected]. The station would be the closest rest �� stop on the interplanetary superhigh- �� way. From there cargo could be sent in slow but energy-efficient, low-thrust freighters, whereas astronauts would travel in higher-speed vehicles. Space- craft leaving the facility could reach any �� point on the lunar surface within hours, �� making it a perfect way station for the return of people to the Moon. This gateway would also be an excellent point of departure and arrival for conven- �� tional interplanetary flights to Mars, the asteroids and the outer solar system. Natural paths for journeying between planets without using fuel exist too, but Figure 9. Exploration of Jupiter’s icy moons could benefit from a cleverly designed trajectory. they require thousands of years to get A probe could, for example, enter the Jovian system along an inbound tube (outer green swath you to your destination. Only asteroids, at left) that carried it toward Jupiter’s moon Ganymede, which it would orbit briefly before comets and Martian meteorites (rocks following an outbound tube (orange) that conveyed it into an orbit around Jupiter that was smaller than Ganymede’s. The probe would then hop to an inbound tube toward Jupiter’s blasted off Mars that later landed on moon Europa (inner green swath), which it would then orbit for a significant time (right). Earth) have the patience for that. Future space telescopes destined for

et had followed an interplanetary sub- Lagrange-point mission, the Interna- deployment near Earth’s L1 or L2 points way tunnel linking distantly separated tional Sun-Earth Explorer 3. could be assembled at this station and regions of space. Portions of these pas- Genesis took a low-energy passage- conveyed to their final destinations us- sageways can run into the planet itself. way to its halo orbit, stayed there while ing very little fuel. And when these in- Oterma’s cousin comet Shoemaker- collecting samples and then found its struments require servicing, they could Levy 9 may have been traveling in just way home on another low-energy path be returned to the vicinity of the station,

such a tube when it broke up and col- that looped by L2. Using their knowl- again without costing much fuel. lided with Jupiter in 1994. edge of the tubes, lead mission design- But the exploitation of low-energy er Lo, along with Purdue University’s passageways is in no way limited to Hitchhiker’s Guide Kathleen C. Howell and Brian Barden near-Earth space. I’m part of an inter- If you could hitch a ride on Oterma or (who was then Howell’s student), national team (one that includes Koon, one of the other natural objects that found a way for Genesis to achieve Marsden, Lo, Gerard Gómez of the travels the tube-highway between the this exotic trajectory using hardly any University of Barcelona and Josep Mas- planets, you could get around the so- fuel. That feat created a great deal of demont of the Technical University of lar system for free. But why wait for interest in both the astronautical and Catalonia, also in Barcelona) that has the right asteroid to come by? All you mathematical communities. proposed a new class of space missions. need to do is direct your spacecraft In particular, the work on Genesis in- Our idea is that a single spacecraft could into one of these celestial conduits. spired Lo and me to explore the dynam- orbit several of the moons of any one Traveling in these passageways would ics of Earth’s neighborhood in a deeper of the outer planets, allowing for long- slash the amount of fuel required to way. We recognized that Lagrange duration observations. For example, a

explore the solar system. The place to points L1 and L2 in both the Sun-Earth multi-moon orbiter could explore Ju- start to look for such opportunities is and Earth-Moon systems are important piter’s planet-sized and likely water- right around Earth. hubs and destinations. Fortunately, the bearing moons—Callisto, Ganymede Because it has an open view of the tubes connecting the neighborhoods of and Europa—one after the other, tak-

cosmos, Earth’s L2 Lagrange point is these four Lagrange points are such that ing a path that uses a technologically well suited for deep-space telescopes, they sometimes intersect one another. feasible amount of fuel. NASA had been

whereas the region around L1, because Once each month or so, halo orbits considering just such a project, dubbed of its unobstructed view of the Sun, around the Moon’s L1 and L2 Lagrange the Jupiter Icy Moons Orbiter, which is a good place to put instruments for points connect to halo orbits around the would exploit linkages among the low-

doing solar science. Indeed, part of Earth’s L1 or L2 points via low-fuel, or energy tubes of Jupiter and its moons, the reason that the Genesis mission even fuel-free, pathways. The implica- but funding for that mission was slashed was feasible was its special “halo or- tions of this fortuitous arrangement for last year, and its prospects are in doubt.

bit” around Earth’s L1 Lagrange point. the exploration and development of the As viewed from the vantage point of solar system are enormous. From Atoms to Galaxies someone on Earth, Genesis moved in a Lo and I, along with colleagues at The growing understanding of the re- halo around the Sun. Such orbits were NASA, have championed the idea that a stricted three-body problem and the dy- originally named for lunar halo orbits permanent space station be established namics associated with Lagrange points

by their discoverer in the 1960s, Robert at the lunar L1 Lagrange point to serve will surely aid in the exploration and Farquhar of Johns Hopkins Universi- as a transportation hub, one that could development of space. But it turns out ty’s Applied Physics Laboratory, who help considerably in advancing space- that the idea of low-energy passageways was the driving force behind the first faring activities beyond low-Earth orbit. has broader application. That realiza-

© 2006 Sigma Xi, The Scientific Research Society. Reproduction 236 American Scientist, Volume 94 with permission only. Contact [email protected]. tion began in 2000, when Charles Jaffé, a chemist at West Virginia University, observed that under proper experimental conditions, the paths taken by valence electrons in Rydberg atoms (whose valence electrons orbit far from an ionized atomic core) look a lot like the trajectory of the Genesis probe. And it indeed turns out that when subjected to electric and magnetic fields that are perpen- dicular, Rydberg electrons also follow tubular pathways. Jaffé teamed up with me, Marsden, Lo, Turgay Uzer of the Georgia Institute of Technology and Da- vid Farrelly of Utah State University to apply techniques from statistical chemistry to study the fate of Martian material shot into space as a result of an impact on that planet. This work was the first application of a well-known technique from chemistry to celestial mechanics. This cross-fertilization has gone in the other direction as well. In collabo- ration with Koon, Marsden, Tomohiro Yanao of Caltech, Frederic Gabern of the University of Barcelona and a group headed by Michael Dellnitz of the University of Paderborn in Ger- many and Oliver Junge at the Munich University of Technology, I’ve been working to develop mathematical and Figure 10. Stars stream outward from the Tadpole Galaxy (Arp 188) along a tubelike channel computational foundations of a reac- that stretches for some 280,000 light-years. This conduit (the galactic equivalent of the tubes tion-rate theory that overcomes some making up the interplanetary transport network) arose through gravitational interaction with of the classical difficulties encountered a compact galaxy that can now be seen lurking behind one of the Tadpole’s spiral arms. (Cour- in chemistry. This work was inspired tesy of ACS Science & Engineering Team and NASA.) by computations of transport in the solar system along tubes and related (such as the “Mice” galaxies) show Bibliography geometrical techniques. It is the un- tubelike conduits connecting them. Conley, C. C. 1968. Low energy transit or- derlying mathematics, of course, that Although they have not been chart- bits in the restricted three-body prob- provides the link between chemistry ed yet, one would expect that similar lem, SIAM Journal on Applied Mathematics 16:732–746. and planetary-system dynamics. tubes connect the solar system with Fukushige, T., and D. C. Heggie. 2000. The Tubes are known to govern structure neighboring stars. Imagine if one of the time-scale of escape from star clusters. and motion over galactic scales too, as two Voyager probes, which have now Monthly Notices of the Royal Astronomical Toshi Fukushige of the University of left the solar system, has entered a tube Society 318:753–761. Tokyo and Douglas Heggie of the Uni- heading toward a region of force bal- Jaffé, C., S. Ross, M. Lo, J. Marsden, D. Far- versity of Edinburgh have shown that ance between the Sun and, say, Alpha relly and T. Uzer. 2002. Statistical theory of asteroid escape rates. Physical Review Letters tubes related to Lagrange points lead Centauri, which is several light-years 89:011101. to the “evaporation” of small star clus- distant. That spacecraft might get a Marsden, J. E., and S. D. Ross. 2006. New ters in orbit around some galaxies. free ride all the way to another star. methods in celestial mechanics and mission Even more dramatic examples of Even so, at the rate the Voyagers are design. Bulletin of the American Mathematical tube-like structures occur when two going, they wouldn’t reach that desti- Society 43:43–73. galaxies interact strongly. About 420 nation for thousands of years. Howev- Smith. D. L. 2002. Next exit 0.5 million kilome- million light-years away, the galaxy er, in the distant past other stars have ters. Engineering & Science LXV(4):6–15. Arp 188, otherwise known as the Tad- come much closer to the Sun than our pole galaxy, reveals evidence of a brief current nearest neighbors. It is likely but violent episode in its past. The huge that exchange of material between our For relevant Web links, consult this tail stretching out of the Tadpole marks solar system and such wandering stel- issue of American Scientist Online: where stars slipped into tubes connect- lar systems has occurred, the tubes being it with an intruder galaxy, one that ing the invisible channels of exchange. has since moved and is now mostly Fans of Douglas Adams should thus http://www.americanscientist.org/ hidden from view. The Tadpole’s tail take heart: Although it might take a IssueTOC/issue/841 is thus a 280,000-light-year bridge to very long time, hitchhiking around the nowhere, but some other galaxy pairs galaxy may indeed be possible.