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Cassini's Elaborate Orbital Mechanics
jamie found an article at the NY Times about the extreme orbital mechanics gyrations required to extend the Cassini mission at Saturn by seven more years. Here's a graphic of the mission extension, which NASA took two years to arrive at. "The plans are for Cassini to keep working for seven more years, but it currently has only 22 percent of the maneuvering propellant it had when it started. Figuring out how to more than double the duration of the mission with less than a quarter of the fuel is hard. Cassini's orbital mechanics present an astonishingly complex exercise in Keplerian physics and geometry. The enormous array of science objectives and targets — moons, rings, Saturn itself — makes it one of the most complex missions ever flown. ... 'Without Titan,' Mr. Seal [Cassini's mission planning supervisor] said, 'we would go into one orbit around Saturn and be stuck there.' Thus Titan, in the argot of orbital mechanics, is Cassini's 'tour engine.' [T]he final 'reference trajectory' ... now includes 56 passes over Titan, 155 orbits of Saturn in different inclinations, 12 flybys of Enceladus, 5 flybys of other large moons — and final destruction."
delta-V is *always* measured in m/s. It's a change in velocity.
General Relativity: Space-time tells matter where to go; Matter tells space-time what shape to be.
I have worked on this problem with a teacher at A&M that is working on this exact problem. Even in simple cases of move from here to here in more then 1 burn can not be numerically solved with current technology (damn computers too slow). So I worked on apply genetic algorithms and Lambert’s equation to solve for minimum delta V. These calculating become much more complex when you can enter a 3rd body (a moon) into this type of calculations.
I have also talked with people at Johnson Space Center about this and they use programs like Matlab to determine the orbit maneuvers and another program I can't recall offhand for visualizing it.
I had the good fortune to be working on the Galileo mission during its Mission Design phase. Many of the techniques used by the Cassini mission designers were developed for Galileo. Disclamer: I was not on the mission design team.
First of all, the Voyager encounters with Jupiter and Saturn were always when the spacecraft were moving away from the sun. However, during the Galileo satellite tour the mission designers realized that the Galileo spacecraft could encounter Callisto, Ganymede, and Europa when moving away and moving toward Jupiter. Furthermore, the closest approach ("encounter") could be targeted to be either in front of the moon (with respect its orbit around Jupiter) or behind it. These choices allowed the designers a great deal of freedom to use the moons' gravity to shape the spacecraft's orbit. As I understand it, they did not just plan the current encounter to obtain the next encounter, but also the encounter after that.
The ability to use a moon to shape a spacecraft orbit depends on the ratio of the mass of the planet to the mass of the moon (for all practical purposes the spacecraft is massless.) Only Io, Callisto, Ganymede, and Europa are able to provide gravity assists at Jupiter, and only Titan at Saturn.
I spoke to Bob Mitchell, Cassini Project Manager, a few years ago and asked him about this specifically. He told me that while it was true that having to go back to Titan every time to change the orbit was a constraint, it also provided the freedom to send the spacecraft out of the "plane" where the moons orbited. At Jupiter it was necessary to stay in the plane to make multiple visits to all the moons, but since at Saturn you must visit the same moon to change the spacecraft's orbit every time (Titan) there is fewer reasons to stay in the plane. And, as you can see from the orbit diagrams, Cassini has traveled outside of the plane many times.
By and large, unless you need super-precise calculations, you can rely solely on Newtonian physics to do orbital calculations. This makes the problem much easier to tackle computationally. The equations of motion cannot be solved analytically, but discrete simulations can be done to arbitrary accuracy extending out for years and years. Relativistic effects will appear as a small cumulative error, but it's small enough that it would probably require only a little more fuel to correct for.
While Saturn is heavy compared to the Earth, the curvature it produces in spacetime is tiny in the grand scheme of things. Even for calculations where the Sun dominates, relativistic effects can safely be ignored in all but the most exacting situations.
Put it this way: if relativistic effects mattered, then Kepler, Galileo, Newton, and others wouldn't have been able to work out the mathematics of non-relativistic orbital mechanics in the first place. Newtonian orbital mechanics is plenty accurate to predict the motions of the planets and other bodies to many decimal places over long stretches of time.
About the only noticeable orbital relativistic effect that I know of in the solar system is a slight perturbation in the orbit of mercury that only became apparent after we'd been observing it for a few centuries. Relativity also comes into play in GPS, but that has a lot more to do with the precise timing of their radio signals than with their orbits.
Ding. We have a winner. Orbital calculations, and optimizing on burn, delta-v and minimizing fuel consumption is really hard. I took a celestial mechanics course as part of my graduate work in Physics, and while I was really good at analytical solutions where they could be achieved, the aerospace engineers who couldn't solve a closed form integral equation to save their lives, could give outstanding solutions for Hohmann transfer orbits, LEO mechanics solutions, and many harebrained options. I was amazed at their creativity, while I was grinding really hard for closed form analytical solutions.
This is brilliant stuff, and their creativity of minimizing the burns yet extending the mission is way cool.
Suppose you were an idiot and suppose you were a member of Congress