Landing on a comet is a pretty neat trick; an even bigger trick is launching a rocket to meet up with the comet 10 years in the future

Although we’ve only had the technology to make and fly space-probes like Rosetta for thirty or forty years, we’ve understood the maths needed to navigate to a comet for a lot longer. This is because all the basic physics used to plan Rosetta’s mission and steer it to its target was known by the year 1700.

The first thing you need to know when planning a rendezvous with a comet is what its orbit is. To calculate this, you need to use Kepler’s laws of planetary orbits, which were worked out way back in the 1630s by Johannes Kepler, a German astronomer. Kepler realised that planets, and most comets, move in elliptical orbits around the Sun – not in circular paths as was once believed. Kepler knew that orbits were elliptical (egg-shaped), but he couldn’t show why this was as he lacked a theory of gravity.

This piece of the puzzle was supplied by Isaac Newton in the 1680s, who helped to establish the basic laws of motion that govern the movement of everything in the solar system, from the Sun to Rosetta and down to tiny grains of dust. Newton also invented calculus – a powerful mathematical technique that allow us to analyse continuously changing quantities, like the position and velocity of our target comet.

(Actually, to be pedantic for a moment, calculus was independently discovered at the same time by Gottfried Leibniz. Three centuries ago there was a heated argument lasting decades about who truly invented calculus – something that is totally insignificant compared to the wonderful range of problems calculus allows us to solve.)

One of the first problems Newton tackled with his shiny new mathematical toolkit was to work out how a planet would move around a star under the force of gravity. Newton was delighted when his sums confirmed that a make-believe planet would indeed travel in a close elliptical orbit, just as Kepler had found. Flushed with this success, Newton tried again, this time with two planets orbiting the pretend star. To his intense frustration he discovered that with just three objects in his model universe, the calculations broke down and the problem couldn’t be solved exactly. The best Newton could come up with was an approximate solution that needed a great many tedious and repetitive calculations to give approximate positions for the planets.

### How did they do it?

We still can’t solve Newton’s so-called ‘three body problem’ exactly. In fact we now know that it can’t ever be solved exactly, but we do have computers that are very good at the tedious and repetitive calculations you need to make to work out how the Earth, the comet and Rosetta will move relative to the Sun, and these are the calculations, starting from the known orbits of the comet and the Earth that would have been used to plan the mission.

If Rosetta was just an inert lump of matter, like a bullet, that we fired it off into space ten years ago it would never have met up with the comet. But Rosetta isn’t a bullet, it’s a sophisticated spacecraft with thrusters which allow it to be controlled as it flies. Using these thrusters the craft was steered on a complex path until it was in the right place to spend three years waiting for the right moment for the comet to turn up. It didn’t matter that when it was launched the exact position of the comet on the rendezvous day wasn’t known: it was enough to set Rosetta off on its journey to the right part of the Solar System. The exact course needed to meet up with the comet could be worked out later in the mission, when the position and path of the comet was known with enough accuracy.

Photo credit: DLR German Aerospace Center, on Flickr, ESA

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