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These periodic orbits around a model of the Jupiter-Europa system show the varied and intricate possibilities that mission planners must knit together.

To plan convoluted tours like this one, trajectory planners use computer models that meticulously calculate the trajectory one step at a time. The planning takes hundreds of mission requirements into account, and it's bolstered by decades of mathematical research into orbits and how to join them into complicated tours. Mathematicians are now developing tools which they hope can be used to create a more systematic understanding of how orbits relate to one another.

"What we have is the previous computations that we've done, that guide us as we do the current computations. But it's not a complete picture of all the options that we have," said Daniel Scheeres , an aerospace engineer at the University of Colorado, Boulder.

"I think that was my biggest frustration when I was a student," said Dayung Koh, an engineer at NASA's Jet Propulsion Laboratory. "I know these orbits are there, but I don't know why." Given the expense and complexity of missions to the moons of Jupiter and Saturn, not knowing why orbits are where they are is a problem. What if there is a completely different orbit that could get the job done with fewer resources? As Koh said: "Did I find them all? Are there more? I can't tell that."

In 2021, Koh came across a paper that discussed how to grapple with chaotic orbits from the perspective of symplectic geometry, an abstract field of math that is generally far removed from messy real-world details. She started to suspect that symplectic geometry might have the tools she needed to better understand orbits, and she got in touch with Agustin Moreno , the author of the paper. Moreno, then a postdoctoral fellow at Uppsala University in Sweden, was surprised and pleased to hear that someone at NASA was interested in his work. "It was unexpected, but it was also quite interesting and sort of motivating at the same time," he said.

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Hi. Since I graduated from college, I've been trying to find cool new interesting things to do. Creating special effects wins as my favorite.

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