TAB 19

A MODEL FOR CHARGED PARTICLES

It may not be inconsequential that the computer simulation of the Earth to Mars trajectory has some subroutines that approximate the behavior of positive and negative charged particles (recall the equations of motion for gravity and electrostatics are identical except for a constant of proportionality). The Earth "capture" of the spacecraft is very easy to optimize, behaving like two oppositely charged particles. The Mars capture is exceedingly difficult (in theory as well as in practice), and the problem behaves as two like charged particles.

First consider the Earth "capture" routine ( 8 ), reverse integrating from conjunction to Earth. Earth is a large, massive body - like a positron compared to a tiny electron, the spacecraft. It was simple to create a dependable and convergent algorithm to target an Earth parking orbit from conjunction. Earth has a large sphere of influence (SOI) that is easy to target because Earth and the spacecraft track a similar trajectory, thus there is a wide "window of opportunity" that permits large step sizes, every one of which is assured a solution.

The Mars capture routine ( 9 ), a direct integration from conjunction to Mars, is exceedingly difficult by comparison to the Earth "capture" routine. Mars is much smaller than Earth, and their relative behavior in the numerical model is similar to the behavior of two like charged particles. It is hard to target Mars' SOI from conjunction, the "window of opportunity" being small because their orbital trajectories and velocities are so dissimilar. In fact, it was literally not possible to create an algorithm to solve the problem directly. The routine stops the integration near SOI, then literally "skewers" Mars - fixing the conditions until a successful trajectory to a parking orbit around Mars is achieved, then integrating back through time from this point - to SOI, then conjunction.

The difficulty in solving the conjunction-to-Mars problem as likened ( 9 ) to a pedestrian standing at a railroad crossing, waiting for the locomotive to approach - then making a dramatic last minute maneuver to get on board the rapidly moving train. This situation is predicated by their relative velocities as well as the geometry of how the orbits intersect. There is literally just one solution - modeling the approach on a sun/Mars free return loop - out of literally thousands of "good possibilities." This is just the kind of behavior attributed to two similarly charged particles.

The real life solution to this problem is actually more difficult than posed in the algorithm because the simulation is a 2D one in which all bodies move in the same plane. In reality, Earth and Mars orbit in planes inclined at about 3 degrees, so the spacecraft approaches Mars in one orbital plane, and the sun/Mars free return orbit is in another plane - e.g. the Mars orbital plane. So the last second maneuver at the L2 point on Mars' SOI includes a slight course correction as well as a plane change (although ideally this plane change would have been done before, when the Earth/Mars orbital planes intersected so that the spacecraft approaches Mars in the same plane as Mars' orbit). This further accentuates just how hard it is to get a spacecraft into a safe orbit around Mars, something all the space agencies have learned the hard way, having collectively only a 25% success rate in all their Mars missions.

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© 2004 WH Clark