# Space Travel:

## Mathematics Uncovers an Interplanetary Superhighway

#### Tube Intersections

Spacecraft can travel along tubes, but they can also change course onto another tube. A small rocket maneuver will do the trick. But there's a way to do it without using any fuel, using the natural highway interchanges of the interplanetary transport network. The figure at right illustrates a tube intersection near the Moon that occurs between a tube departing an L_{1} orbit and a tube approaching an L_{2} orbit. A trajectory which goes from one tube to another without using any fuel is called a heteroclinic trajectory, which means it goes from one orbit to another naturally. In practice, to get onto a heteroclinic trajectory, spacecraft usually use some fuel to make small navigation corrections because of our imperfect knowledge of their position and speed.

There
are also heteroclinic trajectories which connect tubes of two different systems.
For instance, at right a heteroclinic trajectory is shown which uses the intersection
of a tube departing an L_{2} orbit in the Earth-Moon system and a tube
approaching an L_{2} orbit in the Sun-Earth system. Such a pathway would
be useful for a telescope built at a future Lunar L_{1} Gateway Station
(lower middle) to get to a Sun-Earth Lagrange point orbit where deep space
observations could begin. Tube intersections work both ways, so when these
a telescope required servicing, it could be returned to the vicinity of the
station, again without costing much fuel.

Heteroclinic intersections can seem tricky to find as they involve exact timing: while on the departing tube, you need to be at the right place at the right time (and with the right velocity!) to jump onto an approaching tube. But there are computational ways to find these perfectly-timed trajectories, and the results can be spectacular.

The Jupiter moon system is a good place to test out these ideas since there are four planet-sized moons orbiting Jupiter--Io, Europa, Ganymede, Callisto--just like a mini Solar System. As the moons move at different speeds in their orbits, their tubes are dragged along with them. It's possible to time a spacecraft trajectory so that it jumps from a tube going around one moon to a tube going around another, as shown in the figure.

Using this tube-hopping approach, a single spacecraft could orbit and explore explore Jupiter's moons, one after the other, taking a path that uses a technologically feasible amount of fuel. NASA had been considering just such a project, dubbed the Jupiter Icy Moons Orbiter, which would exploit linkages among the tubes of Jupiter and its moons. Without using the tube approach and heteroclinic trajectories, the fuel requirements for such a mission would be prohibitively high, taking it out of the realm of possibility.