Sunday, February 16, 2003

Lagrange, Euler, Poincaré, Lyapunov, Conley, McGehee & Simó:

That's a mouthful of mathematicians. What they have in common is a trail of discovery that was described in Caltech's latest magazine, Engineering & Science (LXV/4).

It makes sense that there is a minimum energy orbit between two planets that is simply tangent to the two ellipses -- the Hohmann transfer orbit. This is what we have used for most interplanetary missions. You fire your engine to give you a change of velocity at a precise time, and then you coast along in your new orbit, reaching your target later along the new orbit path.

However, there are some other very strange orbits possible that depend on the balance between competing gravitational pulls from two or more bodies. These orbits are not intuitive at all, and are not simple conic sections. The simplest of them describe shapes like the edge of a potato-chip, and they cycle around, well... nothing. These are the "halo orbits" that exist around the Lagrange points (a great animation of WMAP settling on an L2 halo orbit is the bottom one here.) We are already using these orbits for several research spacecraft (ISEE3, SOHO, COBE, WMAP, Genesis), and in fact we did a little early navigating of these tubes in the mad dancing of the Galileo probe between Jupiter's moons.

It turns out that each of these halo orbits has a family of possible insertion and exit orbits: think of them as on- and off-ramps, constrained by physics. If you drew all the possible orbits going in and out of a particular halo, they would describe a tube that passed through the halo. This tube is called a Lyapunov tube, or a 'manifold'.

The final leap of insight in this chain is that Lagrange points exist for any two bodies in the solar system. Therefore there are Lyapunov tubes between all the bodies of the solar system -- and therefore there are orbits between any two bodies that can be taken that require very little fuel. The thing they do require is a lot of time.

This is the rationale for some of the Martian meteorites reaching Earth - they travelled Lyapunov tubes completely at random, and some of those tubes led down to the Earth's surface. This also means that there must be asteroids in some really bizarre orbits - and indeed Oterma fits that bill very nicely.

My thinking of course is outside the solar system -- these Lyapunov tubes exist between stars too, but I suspect that the time involved in a Lyapunov transfer orbit between stars is geological in scale. OK, even astronomical.

I also wonder about those spokes, spiral waves, and braids in Saturn's rings that so puzzled us during the Voyager flyby - are these orbital resonances simply reflecting all the multilayered cross-sections of Lyapunov tubes between Saturn's large herd of moons and moonlets?