23� American�cientist,�olume�4 - PDF document

23� American�cientist,�olume�4 © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. eople often picture the solar system as a cosmic clockwork. And why not? With few exceptions, the planets orbit the �un in near�perfect circles, and the moons orbit their planets in the same manner, all mocing with the famous regularity of the heacens. Tne imagines the gracitational field created by this orderly mechanism to be epual ly regular. Drop a rock or spacecraft somewhere close to the �un, and the object should plummet into the huge haps more slowly, back to �arth. Nature, alas, is not so simple. The underlying complication, of course, is that the �arth is orbiting the �un, not just hocering fixed in space. As a re sult, some cery unintuitice things can happen. A rock let go near our planet could find itself following a complex and chaotic path, perhaps orbiting first the �arth, then the �un, and back again, ocer and ocer for years. Add in the tugging of all the other planets and moons, and the possible routes through space can get enormously complicatediand puite interesting. Incestigators from fields as dicerse as mathematics, chemistry and fluid dynamics hace recently recealed the planetary transport network of criss crossing pathways. These incisible highway lanes, originating near a planet or moon, guide traffic through the solar system. Out unlike the thoroughfares one finds on the ground, the space high ways and their interchanges are dynam ic, with lanes mocing past one another according to the carying geometrical relations between planets and moons. �taggering through this tangled web, comets and asteroids find themselces jumping from one lane to another willy�nilly, getting handed off between plan etsior sometimes running into them. For such pieces of cosmic flotsam, the solar system turns out to be more like a oids (not yet, anyway), mission plan ners do maneucer spacecraft and can direct them to jump from lane to lane on the interplanetary highway in such a way that they can tracel cast dis tances using practically no fuel. Like an island castaway who throws a mes sage�laden bottle into the right cur rent at the right time, the controller can send a spacecraft to gracitational sweet spots that procide natural gateways to more distant destinations. Harnessing this effect to good purpose, specialists can plan fuel�efficient routes, ones that would not otherwise be imaginable or technically feasible. ploit subtle gracitational effects. In stead, spacecraft race puickly to their destinations Ouck Rogers style, using chemical rockets. Olasting around in this way is a straightforward exercise. The person planning the trajectory need only to consider the influence of one celestial body at a time. That is, you can treat the departure from �arth and the arrical at a distant planet each to be interactions between the space craft and one massice body. �imilarly, the transfer from �arth�s general neigh borhood to that of a faraway planet may be worked out by considering the spacecraft and the �un alone. Hence one only has to deal with two bodies cal procedure that produced a more precise solution by taking into account all gracitational and nongracitational influences on the spacecraft.For missions sent out to fly past multiple bodiesisay, �upiter, �aturn, tions on the spacecraft is cery short. �o the patched�conic approach works cery well. Tne drawback with this approach, though, is that planetary flybys end up being cery brief. Another is that fuel be comes a major factor limiting the space craft�s itinerary. A prohibiticely large amount of propellant would be needed, UifA�oufspmbofubszAUsbotppsuATfuxpsl�ome mathematical sophistication allows spacecraft to be maneucered ocer large distances using little or no fuel�hane D. Ross namical systems from the California Institute of Technology in 2��4. �ince that time he has been an N�F Sathematical �ciences Postdoctoral Fellow at the Unicersity of �outhern California, and in Au gust he will join the faculty of �irginia Polytechnic Institute and �tate Unicersity in the Department of �ngineering �cience and Sechanics. Ross�s research interests include the study of spacecraft control and mission design, geometrical methods for engineering systems, and mixing and transport processes. Address: Department of Aerospace and Sechanical �ngineering, Unicersity of �outhern California, RRO 21�, Los Angeles, CA ������11�1. Internet: www.shaneross.com 2��I Say8�une 231 www.americanscientist.org © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. for example, to put on the brakes and insert a craft into orbit around some dis tant planet or moon, obserce for a while and then blast off to the next destina tion. And taking extra fuel for maneu cering means that the scientific payload must be made smaller than would oth erwise be possible. Thus mission plan ners hace to strike a balance between the proposed trajectory and the amount of instrumentation that can be carried. The �upiter�bound �alileo probe and the Apollo lunar lander, for example, began their journeys away from �arth with about half of their masses being made up of fuel.In another category entirely was the �enesis Discocery Sission to sample the solar wind, which used only 5 per cent of its mass for fuel. Launched in 2��1, the �enesis spacecraft flew 1.5 million kilometers toward the �un, where it loitered for two and a half years gathering indicidual atoms of the solar wind, ultimately bringing them back to �arth in 2��4. In an unfortunate mishap, the parachute failed to deploy after re�entry, and the sample�return canister was badly damaged when it struck the ground at high speed. Thank fully, scientists were able salcage some of what was collected, the first extrater restrial material brought back to �arth from deep space since the last of the Apollo landings in 1��2 and the first to be collected from beyond the Soon�s orbit. �enesis completed its journey of more than 3� million kilometers using only a minimal amount of propellant. For comparison: A car with a full tank of gas (also about 5 percent of the cehi cle�s total mass) can go only about 5�� kilometers before it�s time for a refill.�oyages like that of the �enesis spacecraft would hace been inconceic able not long ago, but they are now possible thanks to the better appre ciation of the low�energy passageways that wind between planets and moons. Conceptually, the approach needed to mount such a journey through space is similar to what sailors hace long doneitaking adcantage of ocean cur rents to speed them where they want to go. Ancient mariners often discocered natural currents by noting the motion of driftwood or seaweed being carried with them. To some extent modern space nacigators can do the same, ob sercing the mocement of natural ob jects, namely comets and asteroids.A comet called Tterma is particu larly interesting in this regard. �ar ly in the 2�th century, this icy body circled the �un outside �upiter�s or bit. Then, after passing close to that planet in 1�3� Tterma began to orbit inside �upiter. The two bodies met up again in 1�I3, at which point the comet moced back to the outside of �upiter, where it remains today. Dur ing each of its encounters with �u piter, the comet loosely orbited the planet. That is, for a time Tterma was a captured moon. Figure 1. �paceflight typically repuires the expenditure of considerable puantities of propellant. Out after it blasted off from �arth, the �enesis probe was able to tracel 1.5 million kilometers toward the �un (green portion of the trajectory), which is some four times farther than the Soon�s orbit (gray circle). �enesis then orbited the �arth�s Lagrange point (white cross in foreground) collecting particles of the solar wind for two and a half years before traceling millions of kilometers along a circuitous path (blue) that looped by another Lagrange point, (second white cross), before returning to �arth in �eptember 2��4. Amazingly, �enesis completed this cast trek using hardly any fuel. The probe did so by follow ing one of the many possible low�energy paths through the solar system, routes that hace long serced as natural conduits between planets for asteroids and comets. �ome of these conduits lead to collision with �arth, as the �enesis probe�s path did by design. 232 American�cientist,�olume�4 © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. What made this comet moce along such a strange path? The best way to get a sense of the answer is to simplify the problem and consider the motion of the comet (a relaticely small object) as it is being acted on by the graci tational tug of two massice bodies: in this case, the �un and �upiter. In the study of celestial mechanics, this situation is referred to as the restricted three�body problem (restricted by the repuirement that the third body hace negligible mass compared with the other two). Although the full solu tion to this problem is rather hard to fathom, the key elements can be understood with just a little physical Tubes and FunnelsOuilding on Kepler�s work of the pre cious century, Isaac Newton solced the gracitational two�body problem.The result is a relaticely simple for mula, which can be used to compute elliptical orbits or hyperbolic space craft flybys. Out it turns out that it is much more difficult to determine the path a comet or space probe will fol low when it is under the gracitational influence of two bodies in orbit about 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 solce. Sany scientists hace worked hard on it through the years. Newton tried and got fed up. He wanted to write down an epuation that describes the motion of the third body for all time. Out he failed, and the three�body problem was declared Let�s not gice up so easily. The trick is to not seek a tidy little epuation to describe the motion. Instead, one can proceed by considering the problem from a geometrical point of ciew, seek ing intuitice insight into what the so lutions might look like. For this, it is helpful to think of one of those funnel�shaped contraptions one sees ecery now and then at science museums, of ten called “a gracity well,f by cirtue of the physical analogy with gracitation. A little chute on the side allows you to roll a coin into the decice in such a way that it will roll around the inside of the funnel for puite some time. A marble would tracel in the same way.In a frictionless world, a coin or mar ble would just keep circling around such a funnel, mimicking the orbit of, say, �arth around the �un (if one as sumes that the funnel is standing in for the �un�s gracitational well). Out in the real world, a circling coin or marble suffers some frictional loss. �o ocer time its orbit around the funnel decays, causing the coin or marble to spiral in ward and downward. This tendency is easily obserced and indeed forms the economic basis for many of these decices: They ecentually take your money. Out in watching your change disappear, you�ll learn some things. In particular, you�ll notice that the small er the size of the orbit, the more times the coin goes around in a gicen pe riod. That is to say, smaller orbits hace higher angular frepuencies. The same is true for objects orbiting in space. Sercury,�enus Figure 2. Sission planners hace come to appreciate that in certain cases the best routes for spacecraft are not always the most direct ones. In some instances it may be smarter to take ad cantage of the low�energy pathways connecting key points in space. For example, a spacecraft destined for the surface of the Soon might get there cia one of the lunar or 2 Lagrange points (green path). �uch Lagrange points may also serce as way stations for trips to other planets, as shown in this space “subway map.f (Courtesy of NA�A.) Figure 3. The existence of low�energy passageways through space can be understood on an intuitice lecel by considering the physics displayed by a “gracity well.f These funnel�shaped decices (like the one shown here, located at the Sorehead Planetarium in Chapel Hill, North Carolina) allow coins to circle stably much the same way that a planet orbits the �un. �immy Dorff 2��I Say8�une 233 www.americanscientist.org © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. Imagine now a funnel with three marbles circling in three closely spaced, parallel orbits, one just a bit farther out than the next. Compared with the one in the middle, the marble in the outer orbit would hace to tracel at a slower angular frepuency to be stable; the one in the inner orbit would go around at a higher angular frepuency. The same is true in space, say, for three aster oids orbiting the �un. If the middle one were traceling at one astronomical unit from the �un (15� million kilometers, the �arth��un distance), it would take 3I5 days to make one recolution. The outer asteroid would take slightly lon ger than 3I5 days to make a full circle; the inner asteroid would complete its orbit faster than 3I5 days.This pattern is easy enough to un derstand when you recall what hap pens with one of those coin�and� funnel gizmos. Out now imagine that one of those science�museum decices has a small funnel shape embedded in the main funnel. What is more, let that small funnel circle around the larger one at the same rate that a coin or mar ble would hace moced around at that position. This arrangement mimics the gracitational wells of, say, the �arth (the small funnel) and �un (the large funnel) combined.Consider now what would happen if you shot a marble around in such a way that it circled the central depres sion at the same distance from the cen ter and at the same rate as the small (�arth) funnelibut not too close by. The marble would just circle around nicely for a long while. If it were moc ing at the same angular celocity but positioned farther out (where the surface has a gentler slope), this mar ble would be going too fast to circle around stably and would be flung out ward. Concersely, if it were inside of the �arth�funnel�s orbit, it would be mocing too slowly to support itself against the steep walls of the main fun nel and would be drawn inward. The only place a marble with that particu lar angular frepuency circles properly is at the radius of the �arth�funnel�s orbitior is it?Closer scrutiny of this weird surface will receal two cery special points. Tne lies close to the ridge that con nects the little �arth�funnel to the main �un�funnel. Let�s get there by start ing near the center (deep in the �un�funnel) and mocing in the direction of the �arth�funnel. The surface first rises with its usual steepness, but then it rolls ocerithat is, the slope first becomes less steep, then things flat ten out, then you drop into the �arth�funnel. Oefore getting to the top of the intercening ridge, you�ll pass a point where the slope is just right for balanc ing a marble that is circling around at the same rate as the �arth�funnel. Figure 4. Sarbles circling in a gracity wellior planets circling the �unido so at a rate that depends on the size of their orbits: The smaller the orbit, the larger the angular frepuency repuired to achiece balance (left). A demonstration decice engineered to mimic the gracitational field that arises from a pair of massice objects, say the �un and �arth, would be shaped like a large funnel with a small funnel embedded in it (lower right). Here the small funnel would hace to orbit the large one, just as �arth orbits the �un. A marble traceling around the large funnel at the same angular frepuency could balance at two spots that straddle the small funnel (white crosses beneath marbles)icorresponding to �arth�s and Lagrange points. With care, a marble could be positioned and gicen an initial celocity such that it would then “orbitf such special locations (at least for a limited time), just as �enesis was made to orbit �arth�s Lagrange point (upper right). 234 American�cientist,�olume�4 © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. Remember, you�d normally expect this marble to hace to circle around faster to stay balanced. Out because the slope here is somewhat less steeply inclined than is typically the case for this orbital radius, the marble can circle around just fine. This point of balance for the marble has an epuicalent in space. It�s located 1.5 million kilometers from �arth in the direction of the �un.�etting back to the funnel realm, on the outside of the �arth�funnel there is another special balance point. Recall that on this side, a marble would nor mally orbit at an angular frepuency that is less than that of the �arth�fun nel. A marble that moced with the angular frepuency of the �arth�funnel but positioned farther out, where the walls of the main funnel slope less steeply, would normally be expected to fly outward. Out there is one spot where it won�t do that: just on the outside of the �arth�funnel, where the surface is somewhat steeper than normal. Again, there is an epuicalent balance point in space, located 1.5 million kilometers from �arth in the direction opposite the �un.The 1�th�century mathematician Leonhard �uler discocered these two special points (along with a third). His contemporary �oseph�Louis Lagrange discocered two others, and the fice are now known as Lagrange points. Although each represents a special orbit around the �un, they are called “pointsf because they appear as fixed locations when ciewed in a reference frame that rotates at the same rate that the �arth and �un orbit around their center of mass (a point deep inside the �un). Fice such special spots, des through , exist for ecery pair of massice bodiesithe �un and a planet, a planet and one of its moons, and so on. corresponds to the inner balance point for the marble described aboce (the one located between the �arth�funnel and �un�funnel); cor responds to the outer balance point. and are of direct interest for un derstanding the interplanetary trans port network, because they form key Although and are classified as unstable points, that categorization can be misleading, because spacecraft can stick around these points for long periods of time. Indeed, the delicate interplay of gracitational and rotation al forces allows a spacecraft to moce about these points, “orbitingf or in the rotating frame of reference, ecen though there is no material object there. Although such orbits around a mere point in space appear cery bi zarre, they are, in fact, nothing more than near misses to being exactly on or and mocing at just the right celocity for perfect balance.To understand better, imagine that a space probe was orbiting around the �un close to �arth�s point but just a little bit to the inside of it. Assume too that it was mocing just a little bit faster than it needed to go had it been positioned right on . What�s it go ing to do? Again, cisualizing a marble circling around in a double�funnel ar rangement helps. A marble with the corresponding position and celoc ity to this space probe would start to moce ahead in its orbit around the �un�funnel (ahead compared to ); it would also tend to be flung outward slightly. Out the surface around here has a strange shape. �o as the marble moces slightly ahead and outward, the surface in front of it rises, causing the marble to slow. �oon catches up with it (on the inside). The marble then begins to trail and encounters an other rising surface behind, which acts to speed the marble up and to scoot it toward the �un, just as a wace propels a surfer toward the beach (and often a little sideways). �o the marble ends up pretty much where it began and with about the same celocity. It might successfully “orbitf 2 a few times in this manner before either being flung outward or falling off into the nearby �arth�funnel. A spacecraft positioned near can act similarly. �iewed from the perspec tice of someone on �arth, the craft would appear to orbit �arth�s La grange point for a while and then go shooting off toward �arth or around the �uniall without expending any fuel. �ome of the possible orbits about these two Lagrange points lie in the plane of �arth�s orbit. Tthers, like the one followed by the �enesis probe, are �arthSoon Soon�s orbit Figure I. A spacecraft gicen the proper initial celocity can be sent along a trajectory that would then carry it into orbit around, for example, �arth�s Lagrange point (pale green line). A col lection of similar trajectories (any point on which the spacecraft would hace a specific position and celocity) constitutes one “tubef of the interplanetary transport network (green mesh). A spacecraft on a trajectory inside this tube will pass and head toward the outer solar system (blue line) whereas one on a trajectory to the outside will fly back toward the �un (red line) Figure 5. Fice points of gracitational epui librium exist in the restricted three�body problem, as shown here for the �arth�Soon system. All fice moce with the Soon as it orbits �arth. Lagrange points , and are considered unstable, because an object placed at one of these three points will tend to drift slowly away from it ocer time. The and Lagrange points are stable in the sense that objects placed in their cicinity will naturally tend to remain close by. �arthSoon�s orbit 2��I Say8�une 235 www.americanscientist.org © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. three�dimensional and hace a cariety of spiraling shapes, dipping into and out of the orbital plane of the two mas sice bodies.�urfing Oetween PlanetsAt the end of the 1�th century, the French mathematician Henri Poin caré made significant strides in un derstanding 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 ear dynamics, more generally called chaos theory. It is important to keep in mind that “chaoticf does not mean random. Chaotic�looking paths exist in this problem, but they are necertheless predictable, at least for a while into the future. �o a mission designer with sufficient understanding can take full adcantage of them to work out cari ous low�energy routes through space. Poincaré brought order to the chaos by organizing similar paths into spe cial collections of surfaces, which ex ist in what mathematicians refer to as a “six�dimensional phase space,f one that includes the three dimensions of normal space (say, , and ) and three dimensions for an object�s celocity in each direction.Ouilding on Poincaré�s work, in the late 1�I�s Charles C. Conley (a math ematician then at the Unicersity of Wis consin) discocered a collection of tube�shaped surfaces for objects under the gracitational influence of two mutually orbiting bodies, a result later pursued by Robert P. Sc�ehee, then Conley�s student and now at the Unicersity of Sinnesota. An object located on one of these six�dimensional tubes (which is to say, hacing just the right position and celocity) will naturally be carried to ward or away from a trajectory that or bits about the or Lagrange points 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.�tarting in the mid�1���s, I hace worked with Sartin W. Lo of NA�A�s �et Propulsion Laboratory (�PL) and Wang �ang Koon and �errold �. Sars den of the California Institute of Tech nology to extend this approach. We�ce shown that the important physical property of these tubes is that anything that shifts from an orbit that is inside a planet�s orbit to an orbit that lies out side must pass along them. Like water directed by a hose, the set of possible planet�passing objects is imagined to flow along these tubes, but in six dimensions instead of just three. The comparison with fluids is more than just analogy. Indeed, computational tools originally designed by Francois Lekien of Princeton Unicersity and his co�workers for computing dynamical channels in the ocean hace been used to ascertain low�energy trajectories in the celestial context as well.Computing the configuration of these tubes out farther than Conley or Sc�ehee were able to do, my col leagues and I found that they extend far from their region of origin (the ci cinity of or ) and wind around whatecer two massice bodies are be ing considered, stretching and twisting along the way. Tne can think of there being a gateway region around and another around , with the tubes be ing the passageways in and out of the domain of the planet or moon. Anoth er property of objects traceling along such a tube is that they will moce at their slowest relatice to the nearby planet or moon when in the gateway, which can be thought of as a region of near�epuilibrium, the top of an ener getic hill that objects must climb and ocercome.It turns out that Tterma�s strange path lies along the tubular passage ways connected to �upiter�s and Lagrange pointsialmost as if the com Figure �. �ome tubes of the interplanetary transport network lead objects into orbit around �arth�s and Lagrange points (green trajectories, right), whereas others lead objects away from such orbits (orange). Sission planners can make use of the intersection of these incom ing and outgoing tubes, directing a spacecraft to hop from one tube to another in a way that allows it to tracel between and or into orbits around the �un that can be smaller or larger than that of �arth (left). Howecer, a spacecraft with limited kinetic energy (say, just the amount needed to orbit or ) cannot cisit certain regions (gray) no matter what tube pathway it then follows. �un�upiter�upiter�sorbit Tterma�s trajectory 1���1�1� 1�3� 1�I3 1�1� 1���Tterma�s trajectory �upiter �un Figure �. Comet Tterma followed the interplanetary transport network from an orbit that was outside �upiter�s in 1�1� to an orbit that was inside �upiter�s between 1�3� and 1�I3, when it once again shifted to an outside track. Its curious route through space is shown here in both fixed (left) and rotating (right) reference frames. �un�upiter�upiter�sorbit Tterma�s trajectory 1���1�1� 1�3� 1�I3 1�1� 1���Tterma�s trajectory �upiter �un 23I American�cientist,�olume�4 © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. et had followed an interplanetary sub way tunnel linking distantly separated regions of space. Portions of these pas sageways can run into the planet itself. Tterma�s cousin comet �hoemaker�Lecy � may hace been traceling in just such a tube when it broke up and col lided with �upiter in 1��4. Hitchhiker�s �uideIf you could hitch a ride on Tterma or one of the other natural objects that tracels the tube�highway between the planets, you could get around the so lar system for free. Out why wait for the right asteroid to come by? All you need to do is direct your spacecraft into one of these celestial conduits. Traceling in these passageways would slash the amount of fuel repuired to explore the solar system. The place to start to look for such opportunities is right around �arth. Oecause it has an open ciew of the cosmos, �arth�s Lagrange point is well suited for deep�space telescopes, whereas the region around , because of its unobstructed ciew of the �un, is a good place to put instruments for doing solar science. Indeed, part of the reason that the �enesis mission was feasible was its special “halo or bitf around �arth�s Lagrange point. As ciewed from the cantage point of someone on �arth, �enesis moced in a halo around the �un. �uch orbits were originally named for lunar halo orbits by their discocerer in the 1�I�s, Robert Farpuhar of �ohns Hopkins Unicersi ty�s Applied Physics Laboratory, who was the dricing force behind the first Lagrange�point mission, the Interna tional �un��arth �xplorer 3.�enesis took a low�energy passage way to its halo orbit, stayed there while collecting samples and then found its way home on another low�energy path that looped by . Using their knowl edge of the tubes, lead mission design er Lo, along with Purdue Unicersity�s Kathleen C. Howell and Orian Oarden (who was then Howell�s student), found a way for �enesis to achiece this exotic trajectory using hardly any fuel. That feat created a great deal of interest in both the astronautical and mathematical communities.In particular, the work on �enesis in spired Lo and me to explore the dynam ics of �arth�s neighborhood in a deeper way. We recognized that Lagrange points and in both the �un��arth and �arth�Soon systems are important hubs and destinations. Fortunately, the tubes connecting the neighborhoods of these four Lagrange points are such that they sometimes intersect one another. Tnce each month or so, halo orbits around the Soon�s and Lagrange points connect to halo orbits around the �arth�s or points cia low�fuel, or ecen fuel�free, pathways. The implica tions of this fortuitous arrangement for the exploration and decelopment of the solar system are enormous.Lo and I, along with colleagues at NA�A, hace championed the idea that a permanent space station be established at the lunar Lagrange point to serce as a transportation hub, one that could help considerably in adcancing space faring acticities beyond low��arth orbit. 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 tracel in higher�speed cehicles. �pace craft leacing 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 Soon. This gate way would also be an excellent point of departure and arrical for concen tional interplanetary flights to Sars, the asteroids and the outer solar system. Natural paths for journeying between planets without using fuel exist too, but they repuire thousands of years to get you to your destination. Tnly asteroids, comets and Sartian meteorites (rocks blasted off Sars that later landed on �arth) hace the patience for that.Future space telescopes destined for deployment near �arth�s or points could be assembled at this station and conceyed to their final destinations us ing cery little fuel. And when these in struments repuire sercicing, they could be returned to the cicinity of the station, again without costing much fuel.Out the exploitation of low�energy passageways is in no way limited to near��arth space. I�m part of an inter national team (one that includes Koon, Sarsden, Lo, �erard �ómez of the Unicersity of Oarcelona and �osep Sas demont of the Technical Unicersity of Catalonia, also in Oarcelona) that has proposed a new class of space missions. Tur idea is that a single spacecraft could orbit seceral of the moons of any one of the outer planets, allowing for long�duration obsercations. For example, a multi�moon orbiter could explore �u piter�s planet�sized and likely water� bearing moonsiCallisto, �anymede and �uropaione after the other, tak ing a path that uses a technologically feasible amount of fuel. NA�A had been considering just such a project, dubbed the �upiter Icy Soons Trbiter, which would exploit linkages among the low�energy tubes of �upiter and its moons, but funding for that mission was slashed last year, and its prospects are in doubt.From Atoms to �alaxiesThe growing understanding of the re stricted three�body problem and the dy namics associated with Lagrange points will surely aid in the exploration and decelopment of space. Out it turns out that the idea of low�energy passageways has broader application. That realiza Figure �. �xploration of �upiter�s icy moons could benefit from a clecerly designed trajectory. A probe could, for example, enter the �ocian system along an inbound tube (outer green swath at left) that carried it toward �upiter�s moon �anymede, which it would orbit briefly before following an outbound tube (orange) that conceyed it into an orbit around �upiter that was smaller than �anymede�s. The probe would then hop to an inbound tube toward �upiter�s moon �uropa (inner green swath), which it would then orbit for a significant time (right). �upiter �anymede spacecraft trajectory �uropaspacecraft trajectory 2��I Say8�une 23� www.americanscientist.org © 2��I �igma Xi, The �cientific Research �ociety. Reproduction with permission only. Contact perms@amsci.org. tion began in 2���, when Charles �affé, a chemist at West �irginia Unicersity, ob serced that under proper experimental conditions, the paths taken by calence electrons in Rydberg atoms (whose ca lence electrons orbit far from an ionized atomic core) look a lot like the trajec tory of the �enesis probe. And it indeed turns out that when subjected to electric and magnetic fields that are perpen dicular, Rydberg electrons also follow tubular pathways. �affé teamed up with me, Sarsden, Lo, Turgay Uzer of the �eorgia Institute of Technology and Da cid Farrelly of Utah �tate Unicersity to apply technipues from statistical chemis try to study the fate of Sartian material shot into space as a result of an impact on that planet. This work was the first application of a well�known technipue from chemistry to celestial mechanics.This cross�fertilization has gone in the other direction as well. In collabo ration with Koon, Sarsden, Tomohiro Yanao of Caltech, Frederic �abern of the Unicersity of Oarcelona and a group headed by Sichael Dellnitz of the Unicersity of Paderborn in �er many and Tlicer �unge at the Sunich Unicersity of Technology, I�ce been working to decelop mathematical and computational foundations of a reac tion�rate theory that ocercomes some of the classical difficulties encountered in chemistry. This work was inspired by computations of transport in the solar system along tubes and related geometrical technipues. It is the un derlying mathematics, of course, that procides the link between chemistry and planetary�system dynamics.Tubes are known to gocern structure and motion ocer galactic scales too, as Toshi Fukushige of the Unicersity of Tokyo and Douglas Heggie of the Uni cersity of �dinburgh hace shown that tubes related to Lagrange points lead to the “ecaporationf of small star clus ters in orbit around some galaxies.�cen more dramatic examples of tube�like structures occur when two galaxies interact strongly. About 42� million light�years away, the galaxy Arp 1��, otherwise known as the Tad pole galaxy, receals ecidence of a brief but ciolent episode in its past. The huge tail stretching out of the Tadpole marks where stars slipped into tubes connect ing it with an intruder galaxy, one that has since moced and is now mostly hidden from ciew. The Tadpole�s tail is thus a 2��,����light�year bridge to nowhere, but some other galaxy pairs (such as the “Sicef galaxies) show tubelike conduits connecting them.Although they hace not been chart ed yet, one would expect that similar tubes connect the solar system with neighboring stars. Imagine if one of the two �oyager probes, which hace now left the solar system, has entered a tube heading toward a region of force bal ance between the �un and, say, Alpha Centauri, which is seceral light�years distant. That spacecraft might get a free ride all the way to another star. �cen so, at the rate the �oyagers are going, they wouldn�t reach that desti nation for thousands of years. Howec er, in the distant past other stars hace come much closer to the �un than our current nearest neighbors. It is likely that exchange of material between our solar system and such wandering stel lar systems has occurred, the tubes be ing the incisible channels of exchange. Fans of Douglas Adams should thus take heart: Although it might take a cery long time, hitchhiking around the galaxy may indeed be possible.OibliographyConley, C. C. 1�I�. Low energy transit or bits in the restricted three�body prob lem, �IAS �ournal on Applied Sathematics 1I:�328�4I.Fukushige, T., and D. C. Heggie. 2���. The time�scale of escape from star clusters. Sonthly Notices of the Royal Astronomical �ociety 31�:�538�I1.�affé, C., �. Ross, S. Lo, �. Sarsden, D. Far relly and T. Uzer. 2��2. �tatistical theory of asteroid escape rates. Physical Reciew Letters ��:�111�1.Sarsden, �. �., and �. D. Ross. 2��I. New methods in celestial mechanics and mission design. Oulletin of the American Sathematical �ociety 43:438�3.�mith. D. L. 2��2. Next exit �.5 million kilome ters. �ngineering & �cience LX�(4):I815. Figure 1�. �tars stream outward from the Tadpole �alaxy (Arp 1��) along a tubelike channel that stretches for some 2��,��� light�years. This conduit (the galactic epuicalent of the tubes making up the interplanetary transport network) arose through gracitational interaction with a compact galaxy that can now be seen lurking behind one of the Tadpole�s spiral arms. (Cour tesy of AC� �cience & �ngineering Team and NA�A.) 3or relevant Wel links, consult this issue of American Scientist Online http://www.americanscientist.org/ IssueTOC/issue/841